web.math.rochester.edu › people › faculty › jnei › ... · 2012-01-26 · 2 JOSEPH A. NEISENDORFER 14. Samelson products and Lie identities in groups 43 15. Internal Samelson

Contemporary Mathematics

Homotopy groups with coefficients

Joseph A. Neisendorfer

Abstract. This paper has two goals. It is an expository paper on homotopygroups with coefficients in an abelian group and it contains new results which

correct old errors and omissions in low dimensions. The homotopy groups withcoefficients are functors on the homotopy category of pointed spaces. They

satisfy a universal coefficient theorem, give long exact sequences when applied

to fibrations, and have Hurewicz maps into homology groups with coefficients.When the coefficient group is finitely generated, homotopy group functors

are corepresentable as homotopy classes of maps out of a Peterson space. A

Peterson space is a space with exactly one nonzero integral reduced cohomologygroup which is the coefficient group. Kan and Whitehead showed that Peterson

spaces do not exist for the coefficient group of the rational numbers.

Of course, rational homotopy groups can be defined by tensoring theclassical homotopy groups with the rationals. For many nonfinitely generated

groups, homotopy groups can be defined by a direct limit of homotopy groups

with coefficients from finitely generated subgroups. This depends on havingsufficient functoriality in the Peterson spaces of the coefficient subgroups. Co-

efficient group functoriality fails in the presence of 2-torsion.

Contents

1. Introduction 22. What are homotopy groups with coefficients? 43. Peterson spaces and finitely generated coefficients 54. Long exact homotopy sequences for fibrations of loop spaces 75. Peterson spaces and Moore spaces 76. Strong coefficient functoriality 97. Global exponents, decompositions of smash products, and fake multiples 128. Bocksteins, reductions, and inflations 159. Hurewicz maps 1910. Abelian homotopy groups in dimension 3 2611. Classical and modular Hopf invariants 2812. The Dold-Thom theorem 3213. The fibre of the geometric Hurewicz map and uniqueness of smash

decompositions 34

2000 Mathematics Subject Classification. Primary 55-02, 55P60; Secondary 13-02, 13D07.Key words and phrases. homotopy groups.

c©0000 (copyright holder)

1

2 JOSEPH A. NEISENDORFER

14. Samelson products and Lie identities in groups 4315. Internal Samelson products 4716. Universal models and relative Samelson products 5217. Samelson products over the loops on an H-space 5518. Mod p homotopy Bockstein spectral sequences 5719. Samelson products in Bockstein spectral sequences 6320. Nonexistence of rational Peterson spaces 6621. Nonfinitely generated coefficients 6822. Computations with Hilton-Hopf invariants 7123. Cohomology of some cubic constructions 7624. Nonassociativity in smashes of mod 3 Moore spaces 8125. Associativity in smashes of 3 primary Moore spaces 83References 85

1. Introduction

Let X be a pointed space, A an abelian group, and n a positive integer. Ho-motopy groups of X with coefficients in A, denoted πn(X;A), were introduced inthe thesis of Frank Peterson [25] and further developed in this author’s thesis [21].Both of these were heavily influenced by John Moore.

We can define homotopy groups πn(X;A) with coefficients in any finitely gen-erated abelian group A. But if there is two torsion in A or if n = 3, these groups arenot functors of the coefficient groups. There are intrinsic failures here, especiallyat the prime 2, but also at odd primes in dimension 3. This has two consequences.

When there is two torsion or if n = 3, it is impossible to make a naturaldefinition of homotopy groups with non-finitely generated coefficients. In all othercases, we can define homotopy groups with coefficients πn(X;A) in an arbitraryabelian group A by taking direct limits over all finitely generated subgroups of A.

Failure of functoriality in the coefficient group also leads to the fact that thehomotopy groups may have a larger exponent than the coefficient group. We focuson the case of coefficients in cyclic groups of prime power order pr. These homotopygroups have exponent pr unless pr = 2 in which case the exponent is 4. For the3-dimensional homotopy groups, these exponents are determined by means of amod pr version of the classical Hopf invariant.

The equality of the exponent for the coefficient group and the homotopy group isequivalent to decomposing the smash product of two Peterson spaces into a bouquetof two Peterson spaces. The smash product decomposition leads to the definitionof Samelson products in homotopy groups with coefficients mod pr unless pr = 2.When p is a prime greater than 3, these Samelson products give the structure ofa graded Lie algebra to the homotopy groups of a loop space. When p is 2 or 3,the Jacobi identity may fail. When p = 3, we show for the first time here that itmay fail in mod 3 homotopy groups. This result was proved in the author’s thesis.Unfortunately, in the original version of this paper, I made the mistake of thinkingthat the same argument would show that the Jacobi identity may fail in mod 3r

homotopy, r ≥ 2. Brayton Gray pointed out that the Jacobi identity would be validin mod 3r homotopy, r ≥ 2.

When p = 2, the Samelson products may fail to be anti-symmetric.

HOMOTOPY GROUPS WITH COEFFICIENTS 3

The above decompositions of smash products of Peterson spaces are unique upto composition with Whitehead products. This uniqueness is essential to the proofthat the Samelson products give a graded Lie algebra structure in homotopy groupswith mod pr coefficients, p ≥ 3. The proof of uniqueness given here is different fromthat given in the author’s book [24]. It is a more natural argument, depending onthe Hilton-Milnor theorem and the geometric Hurewicz map.

If p is an odd prime, the mod p homotopy Bockstein spectral sequence of a loopspace Er(ΩX) is in all dimensions ≥ 1 a spectral sequence of abelian groups. Ifp = 2 and r ≥ 2, then it is a spectral sequence of abelian groups in all dimensions.

If p is an odd prime, the mod p homotopy Bockstein spectral sequence is inall dimensions ≥ 1 a spectral sequence of Lie algebras except for the fact that theJacobi identity and a related triple vanishing identity may fail if p = 3.

The mod p homotopy Bockstein spectral sequence Er(ΩX) restores the Liealgebra identities for Samelson products. For example, if p = 3, the Lie identitiesare valid in the mod 3 homotopy Bockstein spectral sequence if r ≥ 3. It was knownthat, if p ≥ 5, the Lie identities are valid for all r ≥ 1.

There is a necessary modification of the some maps related to the homotopyBockstein spectral sequence at the prime 2 or even at odd primes in dimension 3.For some purposes, the multiples of the identity map on a Peterson space must bereplaced by so-called fake multiples. For example, the composition

pr−1 : Z/prZ → Z/pZ → Z/prZ

does not correspond to the geometry of Peterson spaces if p = 2 or if p is anodd prime and the Peterson space is in dimension 3. But this modification doesnot change the form of the final description of the homotopy Bockstein spectralsequence from what was given in the author’s book [24].

The author hopes that the presentation here will be definitive for homotopygroups with odd primary coefficients. It is an abridgment, modification, and cor-rection of the treatment of homotopy groups with coefficients in the author’s book[24]. It is said that mistakes in the design of Islamic prayer rugs are intentionallyintroduced as a show of humility since only Allah can produce perfection. And theJapanese Buddhist aesthetic known as wabi-sabi requires that beauty be imper-fect, impermanent, and incomplete. The author has unintentionally followed thesequidelines in previous works.

The ultimate goal of the author’s thesis [21] was applications to stable homo-topy theory with the result that the theory was confined to dimensions ≥ 4. Laterunstable applications made it desirable to extend the theory to lower dimensions.This was done in the author’s AMS memoir [21]. In the course of writing theauthor’s book, it was discovered that the identification of the terms of the mod phomotopy Bockstein spectral sequence as

Er(X) = image pr−1 : π∗(X;Z/prZ)→ π∗(X;Z/prZ)

is valid for all primes p and for all dimensions where we have groups, that is,in dimensions ≥ 3. But the argument needed to extend to the prime 2 and, indimension 3, to odd primes was complicated enough that it was not included inthe book. This argument, included in this paper, requires the nontrivial use ofHilton-Hopf invariants to study the distributivity laws for homotopy classes.


The treatment here is certainly not definitive for 2 primary coefficients. Muchmore exploration of Samelson products and the Bockstein spectral sequence isneeded in that case.

For the record, these modifications at odd primes do not occur in a low enoughdimension to have any effect on the results in [4, 5, 6, 22, 23].

2. What are homotopy groups with coefficients?

What should be the criteria for πn(X;A) to be called the n−th homotopy groupof X with coefficients in A? First of all, depending on X, there should be somerestrictions on n and A in order that πn(X;A) be defined. No matter what A is,πn(X;A) may be undefined for low values of n. Even if it is defined, it may bemerely a set and not a group, much less an abelian group.

We make no attempt to characterize by axioms homotopy groups with coeffi-cients, but we wish to indicate a list of desirable criteria for such groups. For largeenough n, the criteria for πn(X;A) are:

Functor on the homotopy category: For fixed A, πn(X;A) should be afunctor defined from the homotopy category of pointed spaces to the category ofabelian groups. πn(X;A) should also be a functor of A restricted to some fullsubcategory of abelian groups, this will be called coefficient functoriality.

Universal coefficient sequences: There should be short exact sequences0 → πn(X) ⊗ A → πn(X;A) → Tor(πn−1(X), A) → 0. The maps in the universalcoefficient sequences should be natural transformations for X in the homotopycategory of pointed spaces and for A in some full subcategory of abelian groups.We regard the existence of the universal coefficient exact sequences as essential.

Long exact sequences of fibrations: If Fι−→ E

p−→ B is a fibration se-quence of pointed spaces, there should be natural transformations ∂ : πn(B;A) →πn−1(F,A) on the category of fibration sequences which are also natural for Ain some full subcategory of abelian groups. These maps should yield long exactsequences

. . .p∗−→ πn+1(B;A)

∂−→ πn(F ;A)ι∗−→ πn(E;A)

p∗−→ πn(B;A)∂−→ πn−1(F ;A)

ι∗−→ . . . .

The maps in these long exact sequences should be compatible with the maps in theuniversal coefficient theorem, that is, the usual long exact fibration sequences for πnshould commute with the above when combined with the natural transformations

πn( )→ πn( )⊗A→ πn( ;A) and πn( ;A)→ Tor(πn−1( ), A).

Hurewicz maps: There should be Hurewicz maps

φ : πn(X;A)→ Hn(X;A)

which are natural transformations for pointed spaces X and for A in some fullsubcategory of abelian groups. The Hurewicz maps should be compatible with themaps in the two universal coefficient sequences.

The problems we must address are the following:1) the existence of homotopy groups with coefficients.2) coefficient functoriality.


3) the exponents of homotopy groups with coefficients, that is, does an exponente for the abelian group A, eA = 0, imply that e πn(X;A) = 0 for all pointed spacesX.

4) the existence of Hurewicz maps and the truth of Hurewicz theorems.5) the existence of Samelson products in homotopy groups with coefficients and

the validity of the identities for a graded Lie algebra.

3. Peterson spaces and finitely generated coefficients

If the homotopy groups with coefficients A are corepresentable, that is, if thereis a space Pn(A) such that πn(X;A) = [Pn(A);X]∗, then that space must be aPeterson space:

Definition 3.1. An n-th Peterson space Pn(A) is a pointed space with exactlyone nonzero reduced integral cohomology group, occurring in dimension n, such thatthat group is isomorphic to A, that is,

Hk((Pn(A)) =

A, k = n

0, k 6= n.

Similarly, an n-th Moore space Mn(A) is a pointed space with exactly onenonzero reduced integral homology group, occurring in dimension n, such that thatgroup is isomorphic to A, that is,

Hk((Pn(A)) =

A, k = n

0, k 6= n.

We claim that

Theorem 3.2. If homotopy with coefficients πn(X,A) is corepresentable by aspace Pn(A), that is, if πn(X,A) = [Pn(A), X]∗, then Pn(A) is an n-th Petersonspace.

Proof. The universal coefficient theorem implies that

Hk(P (A)) = [Pn(A),K(A, k)]∗ = πn(K(Z, k);A)

= πn(K(Z, k))⊗A =

A, k = n

0, k 6= n

Unfortunately, we shall see that Peterson spaces do not exist for some abeliangroups, for example, there are no rational Peterson spaces Pn(Q). Hence, homotopywith coefficients πn(X;A) cannot always be corepresentable. Nonetheless, it iscorepresentable if the coefficient group A is a finitely generated abelian group andn ≥ 2. In this case, we use Peterson spaces to show that homotopy groups withcoefficients exist and satisfy the four criteria.

Let A be a finitely generated abelian group and let 0→ F1Ψ−→ F0

ε−→ A→ 0 bea free resolution of abelian groups with

F0 =⊕α

Z, F1 =⊕β

Z

being finite direct sums with α running over an index set of order a and β runningover an index set of order b. Then Ψ is represented by a b × a matrix M with a


columns and b rows of integer entries Mβ,α. Let M∗ be the adjoint a × b matrix.Let

S0 =∨α

S1, S1 =∨β

S1

and let M∗ : S1 → S0 be any map which induces Ψ : F1 → F0 in integral cohomol-ogy in dimension 1.

Let P 2(A) be the cofibre of the map M∗ : S1 → S0 and consider the longcofibation sequence

S0M∗

−−→ S1 → P 2(A)→ ΣS0M∗

−−→ ΣS1 → ΣP 2(A)→ Σ2S0M∗

−−→ Σ2S1 → Σ2P 2(A)→ . . . .

It is clear that the spaces P k(A) = Σk−2P 2(A) are Peterson spaces for k ≥ 2.For a finitely generated abelian group A and a pointed space X, we define

πk(X;A) = [P k(A), X]∗, k ≥ 2.Of course, for a fixed finitely generated abelian group A, πk(X;A) is a functor

on the homotopy category with values in the category of sets if k = 2, in thecategory of groups if k = 3, and in the category of abelian groups if k ≥ 4.

If we map the above long cofibration sequence into X, we get the long exactsequence

Παπ1(X)M←− Πβπ1(X)← π2(X;A)← Παπ2(X)

M←− Πβπ2(X)← π3(X;A)←

Παπ2(X)M←− Πβπ2(X)← π3(X;A)← . . .

where the maps M have degrees on the summands which are compatible with theintegers in the matrix, that is, the compositions

Mβ,α : πk(X)→ Πβπk(X)M−→ Παπk(X)→ πk(X)

are multiplications by the integers Mβ,α.In other words the maps M may be identified with

πk(X)⊗ F11⊗M−−−→ πk(X)⊗ F0.

Hence, we get the short exact sequences of the universal coefficient theorem fork ≥ 2 :

0→ πk(X)⊗A→ πk(X;A)→ Tor(πk−1(X), A)→ 0

where, if k = 2 and π1(X) is not abelian, the Tor just means the kernel of the mapM .

It is well known that a sequence of suspensions gives the long exact sequenceof a fibration.

· · · → π3(F ;A)→ π3(E;A)→ π3(B;A)∂−→

π2(F ;A)→ π2(E;A)→ π2(B;A).

In general, this terminates in an exact sequence of pointed sets. See George White-head’s book [31]. In particular, there is a map from the path fibration on the baseto any other fibration

ΩB → PB → B↓ ↓ ↓ 1BF → E → B

and the connecting homomorphism is given by ∂ : πn+1(X;A) = [ΣPn(A), X]∗ =[Pn(A),ΩX]∗ → [Pn(A), F ]∗ = πn(F ;A).


4. Long exact homotopy sequences for fibrations of loop spaces

Definition 4.1. Suppose ΩX is a loop space. We define π1(ΩX;A) = π1(ΩX)⊗A = π2(X)⊗A.

Suppose Fι−→ E

q−→ B is a fibration sequence. We desire hypotheses which willyield the long exact sequence of a looped fibration down to dimension one, that is,there should be a long exact sequence of groups

· · · → π3(ΩF ;A)→ π3(ΩE;A)→ π3(ΩB;A)∂−→

π2(ΩF ;A)→ π2(ΩE;A)→ π2(ΩB;A)∂−→

π1(ΩF ;A)→ π1(ΩE;A)→ π1(ΩB;A)→ 0.

Theorem 4.2. Suppose that π1(F ), π1(E), π1(B) are abelian groups. There issuch a long exact sequence if either of the following two hypotheses are satisfied:

1) π2(E)→ π2(B) is an epimorphism and the sequence of fundamental groupsis short exact

0→ π1(F )→ π1(E)→ π1(B)→ 0.

2) π2(E)→ π2(B) is an epimorphism and Tor(π1(F ), A) = 0.

Proof. : Consider the middle long exact sequence and the vertical short exactsequences

0 0 0↓ ↓ ↓

π2(F )⊗A → π2(E)⊗A → π2(B)⊗A → 0↓ ↓ ↓

π3(B;A)∂−→ π2(F ;A) → π2(E;A) → π2(B;A)

↓ ↓ ↓0 → Tor(π1(F ), A) → Tor(π1(E), A) → Tor(π1(B), A)

↓ ↓ ↓0 0 0

If the composition ∂ : π3(B;A)∂−→ π2(F ;A) → Tor(π1(F ), A) is zero, then ∂

can be factored as π3(B;A)∂−→ π2(F )⊗ A→ π2(F ;A). This is trivially true under

the second hypotheses. Under the first hypotheses, it follows from the fact that theTor sequence is exact beginning on the left with 0.

Once we know that we can factor ∂ as above, the fact that the desired sequenceis long exact is an immediate consequence.

5. Peterson spaces and Moore spaces

It is convenient in this section to assume that coefficient groups A are finitelygenerated abelian groups and hence that the n−th Peterson spaces Pn(A) can beassumed to be finite complexes.

If A is a free abelian group, then Peterson spaces are the same as Moore spaces,Pn(A) = Mn(A) up to unnatural isomorphism of A. But, if A is a torsion group,there is a dimension shift, Pn(A) = Mn−1(A) up to unnatural isomorphism of A.To rid ourselves of this unnaturality, we need to discuss two kinds of duality.


If F is a finitely generated free abelian group, the Z− dual is F ∗ = Hom(F,Z).There is an unnatural isomorphism β : F ' F ∗ given by choosing a dual basis anda natural isomorphism α : F → F ∗∗ given by α(x)(y) = y(x) for x ∈ F, y ∈ F ∗.

Similarly, if T is a finite abelian group, the Q/Z-dual is T ∗ = Hom(T,Q/Z).There is an unnatural isomorphism β : T ' T ∗ and a natural isomorphism α :T → T ∗∗ given by α(x)(y) = y(x) for x ∈ T, y ∈ T ∗. Both isomorphisms followfrom the fact that they are true for finite cyclic groups, that is, if T = Z/kZ, thenβ(`)(r) = r`

k is an isomorphism and α : T → T ∗∗ is a monomorphism betweencyclic groups of the same order.

Since Q is a torsion free injective abelian group, the exact sequence 0→ Z →Q→ Q/Z → 0 gives an isomorphism Hom(T,Q/Z) ' Ext(T,Z).

Let Mn(A) denote a Moore space with exactly one nonzero reduced integralhomology group isomorphic to A in dimension n. The universal coefficient exactsequence

0→ Ext(Hn−1(X), Z)→ Hn(X)→ Hom(Hn(X), Z)→ 0

and the fact that dualizations are idempotent implies that

Theorem 5.1. Let A be a finitely generated abelian group, then

Pn(A) =

Mn(A∗) if A is free.

Mn−1(A∗) if A is finite..

For example, Pn(Z) = Sn = Mn(Z) and Pn(Z/kZ) = Sn−1∪ken = Mn−1(Z/kZ)are both Moore spaces. This and the next lemma give explicit constructions of Pe-terson spaces.

Lemma 5.2. If A and B are abelian groups, then Pn(A⊕B) = Pn(A)∨Pn(B)in the sense that the right side of the equation is a candidate for the left side.

The homotopy uniqueness of Peterson spaces is addressed in

Theorem 5.3. If A is a finitely generated abelian group and n ≥ 3, then thehomotopy type of the Peterson space Pn(A) is determined by A, provided it is simplyconnected.

Proof. Let A = F ⊕ T where F is free and T is finite.

Hk(Pn(A)) =

T ∗ if k = n− 1,

F ∗ if k = n,

0 if k 6= n− 1 or n.

In dimension n− 1 the Hurewicz map is an isomorphism and in dimension n itis an epimorphism.

We pick cyclic generators eβ for F and cyclic generators eα for T . Then thereis a map ∨

α

Sn−1 ∨∨β

Sn → Pn(A)

which is a homology epimorphism in dimension n−1 and a homology isomorphismin dimension n. We can attach n−cells to get a map∨

α

Pn(Z/kαZ) ∨∨β

Sn → Pn(A)


which is a homology isomorphism in all dimensions.Since n ≥ 3 the spaces are all simply connected and this is a homotopy equiv-

alence.

The following examples show that there must be some restrictions in order thatPeterson spaces have unique homotopy type.

Constructing a fake circle: Let α be an element of the fundamental groupand let β be an element of any homotopy group. Denote the left action of α on βby α ∗ β.

Let ι1 : S1 → S1 ∨ S2 and ι2 : S2 → S1 ∨ S2 be the two standard inclusionsand let γ = ι2 − 2ι1 ∗ (ι2).

Let

X = (S1 ∨ S2) ∪γ e3

be the result obtained by attaching a 3-cell to the bouquet by the map γ. Theinclusion S1 → X is a homology equivalence but not a homotopy equivalence. Tosee this, inspect the universal cover of X to see that π2(X) = Z[ 1

2 ] 6= 0.

Constructing a fake Moore space: Let ι1 : S1 → P 2(Z/2Z) ∨ S2 andι2 : S2 → P 2(Z/2Z) ∨ S2 be the two standard inclusions. As before, let γ =ι2 − 2ι1 ∗ (ι2). and let

Y = (P 2(Z/2Z) ∨ S2) ∪γ e3

be the result obtained by attaching a 3-cell to the bouquet by the map γ. Theinclusion P 2(Z/2Z) → Y is a homology equivalence but not a homotopy equiva-lence. To see this, inspect the universal covers to see that π2(P 2(Z/2Z)) = Z andπ2(Y ) = Z ⊕ Z/3Z.

6. Strong coefficient functoriality

Let A and B be finitely generated abelian groups and let Pn(A) and Pn(B) bePeterson spaces. We already know: If n ≥ 2, these Peterson spaces exist, can beconstructed by means of free resolutions, and, if n ≥ 3 and the Peterson spaces aresimply connected, their homotopy types are uniquely determined. We shall showthat, if n ≥ 4, Peterson spaces give functors from the category of finitely generatedabelian groups with no 2-torsion to the homotopy category of pointed spaces.

Theorem 6.1. If n ≥ 2 and the Peterson spaces Pn(A) and Pn(B) are con-structed from free resolutions, then the assignment of the cohomology class to a mapdefines a surjection Ψ : [Pn(B), Pn(A)]∗ → Hom(A,B), [g] 7→ Ψ(g) = g∗.

Proof. Let

0 → F1d0−→ F0

ε−→ A → 0↓ f1 ↓ f0 ↓ f

0 → G1d0−→ G0

ε−→ B → 0

be a map of resolutions.


The spaces Pn−1(F0), Pn−1(F1), Pn−1(G0), Pn−1(G1) are bouquets of n − 1spheres. Let

Pn−1(G0)d∗0−→ Pn−1(G1)

ι−→ Pn(B)ε∗=q−−−→ ΣPn−1(G0)

Σd∗0−−→ ΣPn−1(G1) → . . .

↓ f∗1 ↓ f∗0 ↓ f∗ ↓ Σf∗1 ↓ Σf∗0

Pn−1(F0)d∗0−→ Pn−1(F1)

ι−→ Pn(A)ε∗=q−−−→ ΣPn−1(F0)

Σd∗0−−→ ΣPn−1(F1) → . . .

be maps of cofibrations sequences whose cohomology maps realize these maps ofresolutions.

Then [f∗] 7→ (f∗)∗ = f shows that Ψ is a surjection.

Theorem 6.2. Let g : Pn(B) → Pn(A) be a map which induces 0 in integralcohomology, that is, Ψ(g) = 0.

a) if B and A are finitely generated free abelian and n ≥ 2, then Ψ : [Pn(B), Pn(A)]∗ →Hom(A,B) is a bijection and g is null homotopic.

b) if B if finite abelian, A is finitely generated free abelian, Pn(A) is simplyconnected, and n ≥ 2,then Ψ : [Pn(B), Pn(A)]∗ = A∗ ⊗ B → Hom(A,B) is abijection and hence g is null homotopic.

c) if B if finitely generated free abelian, Pn(B) is a finite bouquet of n spheres,A is finite abelian, n ≥ 3, and h : Pn(B) → Pn(A) is any map, then h∗ = 0 inintegral cohomology, that is, h = g as above and g factors as

g : Pn(B)g−→ Pn−1(F1)

ι−→ Pn(A).

If n ≥ 4, then 2 [Pn(B), Pn−1(F1)]∗ = 2 [Pn(B), Pn−1(A)]∗ = 0. Hence, if A hasodd order, any map is null homotopic and Ψ : [Pn(B), Pn(A)]∗ → Hom(A,B) is abijection.

d) if B and A are finite abelian, and n ≥ 3, then g factors as

g : Pn(B)q−→ Pn(G0)

g−→ Pn−1(F1)ι−→ Pn(A).

If n ≥ 4, then 2 [Pn(G0), Pn−1(F1)]∗ = 0. Hence, if A or B has odd order, g isnull homotopic and Ψ : [Pn(B), Pn(A)]∗ → Hom(A,B) is a bijection.

Proof. a) Let

B =⊕β

Z, A =⊕α

Z

be free abelian and n ≥ 2. Then

Pn(B) =∨β

Snβ , Pn(A) =∨α

Snα

and hence

[Pn(B), Pn(A)]∗ = [∨β

Snβ ,∨α

Snα]∗ =∏β

[Snβ ,∨α

Snα]∗ = (sincen ≥ 2)

∏β

∏α

[Snβ , Snα]∗ = Hom(A,B).


b) Since Pn(A) is simply connected, Pn(A) =∨α S

nα and

[Pn(B), Pn(A)]∗ = [Pn(B),∨α

Snα]∗ = πn(∨α

Snα;B) =

πn(∨α

Snα)⊗B = A∗ ⊗B Ψ'−−→ Hom(A,B).

The last map is known to be a surjection between finite sets of equal cardinalityand is therefore a bijection. This proves b).

The following lemma will be used in the proofs of parts c) and d).

Lemma 6.3. Let A be a finite group with free resolution 0→ F1 → F0 → A→ 0

and let Pn−1(F0) → Pn−1(F1)ι−→ Pn(A)

q−→ Pn(F0) → Pn(F1) be a cofibrationsequence. If F is the homotopy theoretic fibre of q : Pn(A) → Pn(F0), there is a

factorization of ι : Pn−1(F1)γ−→ F → Pn(A) such that γ : Pn−1(F1) → F is a

2n− 3 equivalence.

This is a consequence of the last section on the Serre exact sequence.

c) Given any map h : Pn(B)→ Pn(A) as above with B finitely generated free

abelian and A finite, the composition q h : Pn(B)ι−→ Pn(A)

q−→ Pn(F0) inducesthe zero map in cohomology since B and F0 are free and A is finite. Hence, q his null homotopic and h factors through the homotopy theoretic fibre F of q, thatis, h : Pn(B) → F → Pn(A). By Lemma 4.3, γ : Pn−1(F1) → F is a 2n − 3equivalence. If n ≥ 3, it follows that n ≤ 2n− 3 and h factors as

Pn(B)h−→ Pn−1(F1)

ι−→ Pn(A).

If n ≥ 4, then [Pn(B), Pn−1(F1)]∗ is a direct sum of copies of πn(Sn−1). Hence,it is a direct sum of copies of Z/2Z, for example, h : Pn(B)→ Pn−1(F1) has order2 and is a suspension.

For any integer k and any suspension ΣY , let k : ΣY → ΣY indicate the k−thmultiple of the identity vis a vis the co-H structure.

Since 2h is represented by h2 = ιh2 = ι0 = 0, where 2 : Pn(B)→ Pn(B)is twice the identity, 2 [Pn(B), Pn−1(A)]∗ = 0.

Since h∗

: [Pn−1(F1), Pn(A)]∗ → [Pn(B), Pn(A)]∗ is a homomorphism of

groups and [Pn−1(F1), Pn(A)]∗ = A∗ ⊗ F1, it follows that h = h∗(ι) has the order

of A. Hence, if A has odd order, then h has both order 2 and odd order, therefore,h is null homotopic.

d) Suppose n ≥ 3 and we are given a map g : Pn(B)→ Pn(A) such that g∗ =0 : A→ B in integral cohomology H∗. Then g∗ = 0 : A∗ → B∗ in integral homologyH∗. Hence, the composition Pn−1(G0) → Pn(B) → Pn(A) is null homotopic andg factors as g = h q,

g : Pn(B)q−→ Pn(G0)

h−→ Pn(A)

and, by part c) above, h factors as h = ι g,

h : Pn(G0)g−→ Pn−1(F1)

ι−→ Pn(A).


Now suppose n ≥ 4. Since 0 = 2 ι g = ι g 2 and q : Pn(B)→ Pn(G0) is thesuspension of q : Pn−1(B)→ Pn−1(G0),

2 g = g 2 = ι g q 2 = ι g 2 q = 0 q = 0,

that is, g has order 2.Similarly, if A has odd order k, then 0 = k g and g = 0. Thus, in this case,

Ψ : [Pn(B), Pn(A)]∗ → Hom(A,B) is a bijection and both groups have exponentk.

If B has odd exponent k, then g has order 2 and order k, hence g = 0.The fact that k g = 0 if B has odd exponent k follows from the corollary

Corollary 6.4. If B is a finite abelian group with odd exponent k and n ≥4, then 0 = k : Pn(B) → Pn(B) and hence πn(X,B) has exponent k, that is,k πn(X;B) = 0, for all spaces X.

This follows from the facts that [Pn(B), Pn(B)]∗ → Hom(B,B) is a bijectionby the earlier portion of part d) and hence both groups have exponent k.

The Hilton-Milnor theorem gives that

[Pn(A), Pn(B) ∨ Pn(C)]∗ = [Pn(A), Pn(B)]∗ × [Pn(A), Pn(C)]∗, n ≥ 4

and hence

Corollary 6.5. Let B and A be finitely generated abelian groups such that Ahas no 2-torsion. If n ≥ 4, then

Ψ : [Pn(B), Pn(A)]∗ → Hom(A,B)

is an isomorphism. This situation is called strong coefficient functoriality.

Notice that strong coefficient functoriality is not the same as coefficient func-toriality which it implies.

Corollary 6.6. For all n ≥ 4, there is a functor Pn(A) from the category offinitely generated abelian groups A with no 2-torsion to the homotopy category ofspaces such that, for every homomorphism of such abelian groups f : A → B, themap Pn(f) : Pn(B)→ Pn(A) induces f = Pn(f)∗ in integral cohomology. Hence,the groups (or pointed sets) πn(X;A) are well defined functors of both the spacesX and the finitely generated groups A with no 2-torsion.

7. Global exponents, decompositions of smash products, and fakemultiples

Let p be a prime. If n ≥ 3 and pr 6= 2, we claim that the groups πn(X;Z/prZ)are all annihilated by pr. If p is odd and n ≥ 4, this is an immediate consequenceof strong coefficient functoriality. But if n = 3, it requires the use of modular Hopfinvariants which come in a later section.

Let ` be an integer and, for n ≥ 3, let ` : Pn(Z/kZ) → Pn(Z/kZ) alsodenote the multiple of the identity defined by the suspension structure, that is,` : Pn(Z/kZ) → Pn(Z/kZ) is defined by `(< t, x >) =< `t, x > for t ∈ I.This map is called ` times the identity and induces multiplication by ` on thehomotopy groups πn(X;Z/kZ) when these are abelian groups. We sometimes callthese maps true multiples to distinguish them from what we call fake multiples,


that is, maps which induce multiplication by ` in integral cohomology but whichare not homotopic to the true multiples. For example, this occurs for all n ≥ 3 if kis odd or for all n ≥ 4 if p = 2.

Of course, the range of validity for n is increased to one less if X is a loop space,that is, if k is odd, there are no fake multiples on π3(ΩY ;Z/kZ) = π4(Y ;Z/kZ). Ifthese are just groups, then ` induces the `−th power map.

Clearly, a true multiple ` induces multiplication ` : Z/kZ → Z/kZ on theintegral cohomology groups.

Theorem 7.1. If p is a prime and pr > 2, then the multiple pr : P 3(Z/prZ)→P 3(Z/prZ) is null homotopic.

Proof. We assume here knowledge of the modular Hopf invariants which willbe introduced later. The modular Hopf invariant H(f) is the integer mod pr de-fined for self maps f : P 3(Z/prZ) → P 3(Z/prZ) which induce zero in mod pr

reduced homology. If Cf is the mapping cone and if u ∈ H2(Cf ;Z/prZ) ande ∈ H4(Cf ;Z/prZ) are generators then H(f) is defined by the cup product

u ∪ u = H(f)e.

Two properties of H(f) are that it is zero if and only if f is null homotopic andthat it is zero if f is a suspension.

Since the induced map pr : H3(P 3(Z/prZ);Z) = Z/prZ → Z/prZ = H3(P 3(Z/prZ) :Z) is zero in integral cohomology, it is sufficient to check that the modular Hopfinvariant is zero, H(pr) = 0 ∈ Z/prZ.

The map S1 pr−→ S1 defines the multiple pr = pr ∧ 1 : S1 ∧ P 2(Z/prZ) →S1 ∧ P 2(Z/prZ). If

S1 pr−→ S1 → P 2(Z/prZ)

is a cofibration sequence, then so is the smash of it with P 2(Z/prZ),

S1 ∧ P 2(Z/prZ)pr−→ S1 ∧ P 2(Z/prZ)→ P 2(Z/prZ) ∧ P 2(Z/prZ).

Hence, P 2(Z/prZ)∧P 2(Z/prZ) is the mapping cone of pr : P 3(Z/prZ)→ P 3(Z/prZ).Let e1 and e2 be 1 and 2 dimensional cochains dual to the cells of P 2(Z/prZ)

and recall Steenrod’s computation [28] of the cup product,

e1 ∪ e1 =pr(pr + 1)

2e2.

Let

u = e1 ⊗ e1 ∈ H2(P 2(Z/prZ) ∧ P 2(Z/prZ);Z/prZ),

e = e2 ⊗ e2 ∈ H4(P 2(Z/prZ) ∧ P 2(Z/prZ);Z/prZ)

be generators. Then

u ∪ u = −p2r(pr + 1)2

4e.

Hence, H(pr) = −p2r(pr+1)2

4 which is zero in Z/prZ if pr > 2.

Corollary 7.2. Let n, m ≥ 2. If pr > 2, then there is a decomposition of thesmash into a bouquet,

Pn(Z/prZ) ∧ Pm(Z/prZ) ' Pn+m−1(Z/prZ) ∨ Pn+m(Z/prZ).

If pr = 2, then there is no such decomposition for any such m, n ≥ 2.


Proof. If pr > 2, then the mapping cone of the null homotopic map pr :P 3(Z/prZ)→ P 3(Z/prZ) is

P 2(Z/prZ) ∧ P 2(Z/prZ) ' P 3(Z/prZ) ∨ P 4(Z/prZ).

We just suspend this to get the decomposition into a bouquet for any n ,m ≥ 2.On the other hand, if pr = 2, then we have a nontrivial square e = u∪u = Sq2u

in the mod 2 cohomology of P 2(Z/2Z)∧P 2(Z/2Z). Since Steenrod operations arestable, this nontrivial Sq2 will never vanish via suspension and there can be nobouquet decomposition.

For the localization away from 2 of any 2-cell complex P = S2m−1∪g e2n, Bray-ton Gray [9] used the action of the cyclic group of order 2 to construct homotopyidempotents and hence to give a decomposition of spaces

ΣP ∧ P ' Σ2mP ∨ Σ2n+1P.

This decomposition generalizes the above decomposition of the smash productPn(pr)∧Pm(pr). Away from the primes 2 and 3, both decompositions are commu-tative and associative as needed in section 15 for the theory of internal Samelsonproducts. The associativity requires looking at the action of the symmetric groupon the 3-fold smash. But, if n = m = 2, our decomposition does not require asuspension.

Since the decomposition of the smash product is equivalent to the multiple ofthe identity being null homotopic, we have

Corollary 7.3. If pr > 2 and n ≥ 3, the groups πn(X;Z/2rZ) have exponentpr. But the groups πn(Pn(Z/2Z);Z/2Z) = Z/4Z.

We now introduce useful maps ` : Pn(Z/kZ) → Pn(Z/kZ), n ≥ 2 whichinduce ` : Z/kZ → Z/kZ in integral cohomology and thus look like multiples ofthe identity.

Write P 2(Z/kZ) = S1 ∪k C(S1) where

S1 = z ∈ C||z| = 1, k(z) = zk

is the circle of unit complex numbers and

C(S1) =I × S1

0× S1 ∪ I × 1, < 1, z >' zk

is the attachment of the reduced cone.

Definition 7.4. The fake `−th multiple of the identity is the map

` : P 2(Z/kZ)→ P 2(Z/kZ), z 7→ z`, < t, z >7→< t, z` > .

In general, ` = Σn−2` : Pn(Z/kZ)→ Pn(Z/kZ) is the suspension.

It is interesting to compare the true multiples of the identity with the fakemultiples of the identity.

1) The map ` : Pn(Z/kZ) → Pn(Z/kZ) is defined for n ≥ 2. The map` : Pn(Z/kZ) → Pn(Z/kZ) is defined only for n ≥ 3. The map ` is a suspensionfor n = 3. If n = 3 and pr = 2, ` = 2 has a nontrivial modular Hopf invariant,hence, it is not a suspension.

2) Both ` and ` induce ` : Z/kZ → Z/kZ in integral cohomology, thus theyinduce the same maps in homology and cohomology with any coefficients.

3) If k is odd and n ≥ 4, then the maps ` and ` are homotopic.


4) Both maps satisfy the composition laws ` k = `k and ` k = `k.

We claim

Lemma 7.5. The fake multiples of the identity k : Pn(Z/kZ)→ Pn(Z/kZ) arenull homotopic.

Proof. It is sufficient to check it when n = 2. Consider the general situationof maps of mapping cones given by a commutative diagram

Xf−→ Y → Y ∪f CX

↓ a ↓ b ↓ b ∪ CaW

g−→ Z → Z ∪g CWSuppose we have a map ` : Y →W such that a = ` f and b = g `.Then we have a factorization

b ∪ Ca : Y ∪f CX`∪Ca−−−→ CW → Z ∪g CW

and hence b ∪ Ca is null homotopic.In our case, we choose f = g = a = b = k, ` = 1.

Since the fake mulitples ` and the actual multiples ` induce the same maps inintegral cohomology, ` = ` + α where α : Pn(Z/kZ) → Pn(Z/kZ) induces zero inintegral cohomology. But, in fact more is true:

Theorem 7.6. Let p be a prime. Then for all j ≥ 1, there exist maps δj :P 3(Z/prZ)→ P 3(Z/prZ) which induce zero in integral cohomology such that

pj = pj + δj pj = (1 + δj) pj .

Remark 7.7. The above theorem will be proved in the section on Hilton-Hopfinvariants. Note that each map 1+δj is a cohomology isomorphism and therefore ahomotopy equivalence. Thus, up to composition with a homotopy equivalence, thefake multiples pj = pj and the actual multiples pj are the same. By suspension,this is true for Peterson spaces of all dimensions. Hence, their dual maps

(pj)∗, (pj)∗ = (pj)∗ (1 + δj)∗ : πn(X;Z/prZ)→ πn(X;Z/prZ)

have the same images, which are exactly the elements which are divisible by pj .

8. Bocksteins, reductions, and inflations

Let k and ` be nonzero integers. The canonical maps of exact sequences

0 → Z`−→ Z

ρ−→ Z/`Z → 0↓ ρ ↓ ρ ↓ 1

0 → Z/kZη−→ Z/k`Z

ρ−→ Z/`Z → 0

ρ(1) = 1, η(1) = `.

suggest important maps of coefficient groups. The maps ρ are called reductionmaps and the maps η are called inflation maps.

In this section, we introduce canonical maps between 2-dimensional Petersonspaces which induce these maps in integral cohomology and which make the cor-responding dual diagrams of Peterson spaces commute exactly, not just up to ho-motopy. The maps between higher dimensional Peterson spaces are defined by


suspending these maps. The maps between 2-dimensional Peterson spaces are thefollowing:.

ρ : P 2(Z/kZ)→ S2

ρ : P 2(Z/kZ)→ P 2(Z/k`Z)η : P 2(Z/k`Z)→ P 2(Z/kZ)

To make these definitions we use the same models for Peterson spaces that we usedfor defining fake multiples of the identity:

P 2(Z/kZ) = S1 ∪k C(S1), k : S1 → S1, k(z) = zk.

Definition 8.1. 1) The coreduction map ρ : P 2(Z/kZ) → S2 is the quotientmap q which pinches the bottom circle to a point, q(z) = ∗ for all z ∈ S1.

2) The coreduction map ρ : P 2(Z/kZ)→ P 2(Z/k`Z) is the map

z 7→ z`, < t, z >7→< t, z >

3) The coinflation map η : P 2(Z/k`Z)→ P 2(Z/kZ) is the map

z 7→ z, < t, z >7→< t, z` >

It is easy to check that these maps are compatible with the identifications. Thecoreduction and coinflation maps give commutative diagrams of maps of cofibrationsequences

S1 k−→ S1 ι=β−−→ P 2(Z/kZ)q=ρ−−→ S2 k−→ . . .

↓ 1 ↓ ` ↓ ρ ↓ 1

S1 k`−→ S1 ι=β−−→ P 2(Z/k`Z)q=ρ−−→ S2 k−→ . . .

S1 k`−→ S1 ι=β−−→ P 2(Z/k`Z)q=ρ−−→ S2 k`−→ . . .

↓ ` ↓ 1 ↓ η ↓ `

S1 k−→ S1 ι=β−−→ P 2(Z/kZ)q=ρ−−→ S2 k−→ . . .

Definition 8.2. The maps ι = β : S1 → P 2(Z/kZ) are called coBocksteins.As with the coreductions and coinflations they are defined on higher dimensionalPeterson spaces by suspension.

The diagrams above help in identifying the following compositions below. Thecompositions 3) and 4) below are equal to the fake multiples of the identity ` inthe previous section. After suspension, they are equal to the true multiples of theidentity only if n ≥ 4 and if p is odd.

Lemma 8.3. 1)

ρ = ρ ρ : P 2(Z/kZ)ρ−→ P 2(Z/k`Z)

ρ−→ P 2(Z/k`mZ)

2)

η = η η : P 2(Z/k`mZ)η−→ P 2(Z/Z/k`Z)

ρ−→ P 2(Z/k)

3)

` = η ρ : P 2(Z/k`Z)η−→ P 2(Z/kZ)

ρ−→ P 2(Z/k`Z)

4)

` = η ρ : P 2(Z/kZ)ρ−→ P 2(Z/Z/k`Z)

η−→ P 2(Z/kZ)


Lemma 8.4. Applying P 2 to the diagram which began this section, reversingarrows, replacing reductions by coreductions, inflations by coinflations, and multi-plications by fake multiples produces a strictly commutative diagram.

Remark 8.5. The coreduction maps

ρ = q : P 2(Z/kZ)→ S2

pinch the bottom cell to a point and induce the reduction maps

ρ : π2(X)→ π2(X;Z/prZ).

There is a factorization

ρ = ρ ρ : P 2(Z/kZ)→ P 2(Z/k`Z)→ S2

and a consequent factorization

ρ = ρ ρ : π2(X)→ π2(X;Z/k`Z)→ π2(X;Z/kZ).

The fact that the fake multiples ` : P 2(Z/`Z)→ P 2(Z/`Z) are null homotopicyields

Theorem 8.6. The compositions

P 2(Z/kZ)ρ−→ P 2(Z/k`Z)

η−→ P 2(Z/`Z)

are null homotopic.

Proof. Let d = (k, `) = the greatest common divisor. The above compositionis identical to the composition

P 2(Z/kZ)η−→ P 2(Z/dZ)

d−→ P 2(Z/dZ)ρ−→ P 2(Z/`Z).

The point of the above theorem is that short exact sequences of abelian groupsshould correspond to cofibration sequences. This is rather close to being true.

Consider the natural maps

P 2(Z/kZ)ρ−→ P 2(Z/k`Z)

ι−→ Cρj−→ P 3(Z/kZ)

ρ−→ P 3(Z/k`Z)↓ 1 ↓ 1 ↓ ↓ ↓ 1

P 2(Z/kZ)ρ−→ P 2(Z/k`Z)

η−→ P 2(Z/`Z)ι−→ Cη

j−→ P 3(Z/k`Z)

The map Cρ → P 2(Z/`Z) induces an isomorphism in integral cohomology (there-fore in integral homology) and an isomorphism of fundamental groups. Perhaps itis not a homotopy equivalence but any suspension of it is. Hence, if Y is a loop

space, [P 2(Z/`Z), Y ]∗∼=−→ [Cρ, Y ]∗ is an isomorphism and, from the point of view

of maps into Y ,

P 2(Z/kZ)ρ−→ P 2(Z/k`Z)

η−→ P 2(Z/`Z)β−→

P 3(Z/kZ)ρ−→ P 3(Z/k`Z)

η−→ P 3(Z/`Z)→ . . .

behaves just like a cofibration sequence.The map P 3(Z/kZ) → Cη induces an isomorphism in integral cohomology.

Since the spaces are simply connected, it is a homotopy equivalence. The mapβ : P 2(Z/`Z)→ Cη = P 3(Z/kZ) is called the coBockstein.


For dimension reasons it is clear that we have a factorization of the coBocksteinas:

β = β ρ : P 2(Z/`Z)→ S2 → P 3(Z/kZ)

where the second map is the inclusion of the bottom cell and is therefore a co-H-map.

The coBockstein and its suspensions induce the Bockstein homomorphisms

β = ρ β : πn(X;Z/kZ)→ πn−1(X)→ πn−1(X;Z/`Z).

We get

Theorem 8.7. For n ≥ 3, the coreduction and coinflation maps form a cofi-bration sequence

Pn−1(Z/k`Z)η−→ Pn−1(Z/`Z)

β−→ Pn(Z/kZ)ρ−→ Pn(Z/k`Z)

η−→ Pn(Z/`Z)

Remark 8.8. Suppose Y is any space. The cofibration sequence above yieldsthe middle horizontal Bockstein sequence which is exact in the first three terms onthe left. The bottom sequence is exact by the properties of Tor:

0 0 0↓ ↓ ↓

π2(Y )⊗ Z/`Z → π2(Y )⊗ Z/k`Z → π2(Y )⊗ Z/kZ → 0↓ ↓ ↓

π3(Y ;Z/kZ)β−→ π2(Y ;Z/`Z) → π2(Y ;Z/k`Z) → π2(Y ;Z/kZ)

↓ ↓ ↓0 → Tor(π1(Y ), Z/`Z) → Tor(π1(Y ), Z/k`Z) → Tor(π1(Y ), Z/kZ)

↓ ↓ ↓0 0 0

The connecting homomorphism or Bockstein factors as β : π3(Y ;Z/kZ) →π2(Y )⊗Z/`Z → π2(Y ;Z/`Z). The map π3(Y ;Z/kZ)→ π2(Y )⊗Z/`Z is a homo-morphism since S2 → P 3(Z/kZ) is a co-H-map.

Hence, we can splice the sequences to get a long exact Bockstein sequence

· · · → π3(Y ;Z/kZ)→ π2(Y )⊗ Z/`Z → π2(Y )⊗ Z/k`Z → π2(Y )⊗ Z/kZ → 0.

In particular, if X is a loop space, the above can be written as the long exactBockstein sequence

· · · → π2(X;Z/kZ)→ π1(X;Z/`Z)→ π1(X,Z/k`Z)→ π1(X;Z/kZ)→ 0

which terminates in 0.

Remark 8.9. Consider the fake multiples ` : P 2(Z/2Z) → P 2(Z/2Z). If ìs even, ` = 2 2k, then ` is null homotopic. If ` is odd, then ` induces iso-morphisms on integral homology and hence also on the fundamental group andintegral cohomology. But ` is not a homotopy equivalence if ` 6= 1 since it iscovered on the universal cover S2 by a degree ` map. To see this, write the pro-jective plane P 2(Z/2Z) as the quotient of the universal cover, that is, the quo-tient of the suspension S2 = ΣS1 = < t, z > | t ∈ [−1, 1], z ∈ S1 by therelations < t, z >∼< −t,−z > . Since ` is odd, the fake multiple ` is given by< t, z > 7→< t, z` >. This lifts to the universal cover by < t, z > 7→< t, z` >, thatis, by ` : S2 → S2.


9. Hurewicz maps

In this section, we introduce the mod k Hurewicz maps and prove the mod kHurewicz theorems, including the results on the equivalence of mod k homologyisomorphisms and mod k homotopy isomorphisms.

Recall that Hn(Pn(Z/kZ), Z/kZ) ' Z/kZ and let en be a generator. ThenHn−1(Pn(Z/kZ), Z/kZ) ' Z/kZ is generated by the Bockstein β(en) associatedto the short exact sequence 0→ Z/kZ → Z/k2Z → Z/kZ → 0.

Definition 9.1. For n ≥ 2, the Hurewicz map φ : πn(X;Z/kZ)→ Hn(X;Z/kZ)is the map defined by φ[f ] = f∗en.

The Hurewicz map is clearly a natural transformation. It is an exercise to checkthat the Hurewicz map commutes with reductions, inflations, and Bocksteins.

Lemma 9.2. For all n ≥ 2, the following diagrams commute

πn+1(X;Z/`Z)β−→ πn(X;Z/kZ)

η−→ πn(X;Z/k`Z)ρ−→ πn(X;Z/`Z)

↓ φ ↓ φ ↓ φ ↓ φHn+1(X;Z/`Z)

β−→ Hn(X;Z/kZ)η−→ Hn(X;Z/k`Z)

ρ−→ Hn(X;Z/`Z)

πn(X)ρ−→ πn(X;Z/`Z)

β−→ πn−1(X)`−→ πn−1(X)

↓ φ ↓ φ ↓ φ ↓ φHn(X)

ρ−→ Hn(X;Z/`Z)β−→ Hn−1(X)

`−→ Hn−1(X)

For n ≥ 3, the Peterson space Pn(Z/kZ) is a suspension and hence the diagonalis given by ∆(en) = en ⊗ 1 + 1⊗ en. Hence,

Lemma 9.3. If n ≥ 3, the mod k Hurewicz map φ : πn(X;Z/kZ)→ Hn(X;Z/kZ)is a homomorphism.

Proof. Given maps f, g : Pn(Z/kZ)→ X, the sum is given by the composi-tion

Pn(Z/kZ)→ Pn(Z/kZ) ∨ Pn(Z/kZ)f∨g−−→ X ∨X → X

and this induces in homology

en 7→ en ⊗ 1 + 1⊗ en 7→ f∗en ⊗ 1 + 1⊗ g∗en 7→ f∗en + g∗en = φ(f) + φ(g).

If X is a group-like space, then the multiplication gives a group structure onπ2(X;Z/kZ). We have

Lemma 9.4. If X is a group-like space, then the mod k Hurewicz map φ :π2(X;Z/kZ)→ H2(X;Z/kZ) satisfies

φ(f + g) = φ(f) + φ(g) +k(k + 1)

2(βφ(f)) ∗ (βφ(g))

where the last term is the Pontrijajgin product. Hence, if k is odd, this Hurewiczmap is a homomorphism.

Proof. The sum in homotopy is the composition

Pn(Z/kZ)∆−→ Pn(Z/kZ)× Pn(Z/kZ)

f×g−−−→ X ×X → µX.


Steenrod’s computation [28] is that ∆∗(e2) = e2⊗1+1+1⊗e2 + k(k+1)2 (βe2⊗βe2).

The result follows.

Remark 9.5. Since the Bockstein factors through the subgroup Tor(π1(X), Z/kZ) ⊂π1(X), it follows that the above mod k Hurewicz map is a homomorphism if k iseven and Tor(π1(X), Z/kZ) is annihilated by k

2 .

Recall that a pointed space X is called nilpotent if the action of the fundamentalgroup on all the homotopy groups is nilpotent.

Let π = π1(X) be the fundamental group and let Z[π] be the group ring of π.Suppose for simplicity that π is abelian. Hence, π acts trivially by conjugation onitself.

For all j ≥ 1, the homotopy group πj(X) is a module over Z[π].Let I be the augmentation ideal, I = kernel ε : Z[π]→ Z, ε(g) = 1∀g ∈ π.

Definition 9.6. X is nilpotent if, for all j ≥ 1, there is a positive integer njsuch that Injπj(X) = 0.

For a nilpotent X, the descending filtration

πj(X) ⊇ Iπj(X) ⊇ I2πj(X) ⊇ . . . Inj−1πj(X) ⊇ Injπj(X) = 0

is finite and terminates at 0. The action on each quotient πj,`(X) = I`πj(X)/I`+1πj(X)is trivial.

The effect of this is that there is a refinement of the fibration sequences in thePostnikov system for X, that is, K(πj,`(X), j)→ Xα → Xα−1 → K(πj,`(X), j+ 1)are all orientable and there is a homotopy equivalence X → lim←Xα.

We adopt the convention that π1(X;Z/kZ) = π1(X)⊗ Z/kZ.

Theorem 9.7 (mod k Hurewicz theorem). Suppose X is a nilpotent space withabelian fundamental group and n ≥ 1. Suppose that Tor(π1(X), Z/kZ) = 0. Ifπ`(X;Z/kZ) = 0 for all ` < n, then π`(X;Z/kZ) = 0 for all ` < n and theHurewicz map φ : πn(X;Z/kZ) → Hn(X;Z/kZ) is an isomorphism. If n ≥ 2,then the Hurewicz map φ : πn+1(X;Z/kZ)→ Hn+1(X;Z/kZ) is an epimorphism.

Remark 9.8. The significant effect of the hypothesis Tor(π1(X), Z/kZ) =0 is that π2(X;Z/kZ) = π2(X) ⊗ Z/kZ is a group and hence that all of thehomotopy groups with coefficients are groups. In addition, the Hurewicz map is ahomomorphism in all dimensions.

Before we begin the proof of the above Hurewicz theorem, we present here aquick summary of the Serre exact sequence. The presentation is heavily influencedby [19].

Let Fι−→ E

p−→ B be an orientable fibration sequence. Suppose that F is r − 1connected and B is s− 1 connected.

Without changing notation we can regard Ep−→ B as an inclusion whenever it

is convenient. For example, we have maps of cofibration sequences

Fι−→ E → E/F → ΣF

Σι−→ ΣE↓ 1 ↓ ↓ ↓ 1E → B → B/E → ΣE


→ ΣE/F → Σ2FΣ2ι−−→ Σ2E . . .

↓ ↓ ↓ 1Σp−−→ ΣB → ΣB/E → Σ2E . . .

The transgression in the Serre spectral sequence for the fibration Ep−→ B is

defined by the relation

τ : H∗B ← H∗E/F → H∗ΣF = H∗−1F.

The Serre spectral sequence of the pair

E2p,q = Hp(B, ∗;HqF ) =⇒ Hp+q(E,F )

shows that the map H∗(E,F ) → H∗(B, ∗) is an r + s equivalence, that is, anisomorphism for ∗ ≤ r + s − 1 and an epimorphism for ∗ = r + s. Thus, thetransgression τ is a well defined map for ∗ ≤ r + s− 1.

The Serre spectral sequence of the fibration gives the Serre exact sequence withthe transgression as the connecting homomorphism

Hr+s−1F → Hr+s−1E → Hr+s−1Bτ−→ Hr+s−2F

→ Hr+s−2E → Hr+s−2Bτ−→ Hr+s−3F →

· · · → H1F → H1E → H1B

With the understanding that H∗(E,F )'−→ H∗B is an isomorphism in the

range ∗ ≤ r+ s− 1, the Serre exact sequence is just the exact sequence of the pair(E,F ). We know that the Hurewicz maps give a map of the long exact sequenceof homotopy groups of a pair to the long exact sequence of homology groups ofthe pair. Since π∗(E,F ) ∼= π∗B for all ∗, the long exact sequence of the homotopygroups of a fibration map to the Serre exact sequence in the range where the Serreexact sequence is valid.

Since E/F → B is an r + s equivalence, the 5-lemma shows that ΣF → B/Eis also an r + s equivalence.

Furthermore, suppose we have a factorization

Fι−→ E → E/F → ΣF

Σι−→ ΣE↓ g ↓ 1 ↓ ↓ h = Σg ↓ 1C → E → B → B/E → ΣE

where the bottom row is a cofibration sequence up to homotopy. Since h is anr + s equivalence in homology and homotopy, it follows that its desuspension g isan r + s− 1 equivalence.

Proof. Let Cn(X) be the mod k Hurewicz conclusion for a space X and fora fixed integer n and let Hn(X) be the mod k Hurewicz hypothesis for a space Xand for a fixed integer n. We want to prove that Hn(X) implies Cn(X) when X isnilpotent with abelian fundamental group.

The proof proceeds by proving the mod k Hurewicz theorem in three successivecases, 1) X = K(π, 1), 2) X = K(A, q), A abelian, q ≥ 2, 3) X = an arbitrarynilpotent space.


First of all, we note that, since the fundamental group π1(X) is abelian, themod k Hurewicz theorem is true for n = 1, H1(X;Z/kZ) = H1(X) ⊗ Z/kZ =π1(X)⊗ Z/kZ = π1 ⊗ Z/kZ, H1(X) implies C1(X)

Case 1. X = K = K(π, 1) :

When p is a prime, Cartan’s calculation [2] is

H∗(K(A, 1);Z/pZ) = E(A⊗ Z/pZ, 1)⊗ Γ(Tor(A,Z/pZ), 2)

where E is an exterior algebra and Γ is a divided power algebra. The Hurewiczhypothesis for n = 2 implies that A ⊗ Z/pZ = 0 and hence H2(K;Z/pZ) =Tor(A,Z/pZ) and H3(K;Z/pZ) = 0 which is the mod p Hurewicz theorem inthis case, H2(K) implies C2(K) when k = p = a prime.

Suppose we know the mod d Hurewicz theorem for n = 2 and for all properdivisors of k. We have a map of long exact Bockstein sequences. (Remark: SinceK is an abelian topological group, π2(K,Z/kZ) is an abelian group and the modk Hurewicz map is a homomorphism.)

π3(K,Z/dZ) → π3(K,Z/kZ) → π3(K,Z/kdZ) →↓ φ ↓ φ ↓ φ

H3(K,Z/dZ) → H3(K,Z/kZ) → H3(K,Z/kdZ) →

π2(K,Z/dZ) → π2(K,Z/kZ) → π2(K,Z/kdZ) → 0↓ φ ↓ φ ↓ φ

H2(K,Z/dZ) → H2(K,Z/kZ) → H2(K,Z/kdZ) → 0

The 5-lemma implies that the mod k Hurewicz theorem is true for K and n = 2,that is, H2(K) implies C2(K) for all k.

Now suppose n ≥ 3. The mod k Hurewicz hypothesis in this case implies thatA ⊗ Z/kZ = Tor(A,Z/kZ) = 0 and that A ⊗ Z/pZ = Tor(A,Z/pZ) = 0 for allprimes dividing k. Hence, H∗(K(A,Z/pZ)) = Z/pZ and thenH∗(K(A, 1);Z/kZ) =0. Since π∗(K(A, 1);Z/kZ) = 0, the mod k Hurewicz theorem is true in this case,Hn(K) implies Cn(K) for all n and k.

Case 2. X = K = K(A, q), q ≥ 2 :

If q = 2, we claim that H2(K;Z/kZ) = A ⊗ Z/kZ and H3(K;Z/kZ) =Tor(A,Z/kZ)). This is the mod k Hurewicz theorem when n = 2, H2(K) impliesC2(K) for any k when q = 2.

If d is a proper divisor of k, the long exact Bockstein sequences associated withthe coefficient sequence

0→ Z/dZ → Z/kZ → Z/k

dZ → 0

show that it is sufficient to prove the above when k = p = a prime. In detail:The Eilenberg-Moore spectral sequence [18, 24] converging to H∗(K;Z/pZ)

has

E2 = TorH∗(K(A,1);Z/pZ)(Z/pZ,Z/pZ) =

TorE(A⊗Z/pZ,1)(Z/pZ,Z/pZ)⊗ TorΓ(Tor(A,Z/pZ),2)(Z/pZ,Z/pZ) =

Γ(A⊗ Z/pZ, 2)⊗H∗(K(B, 3);Z/pZ)


where B is a free Z module such that there is an isomorphism B ⊗ Z/pZ →A. The identification TorΓ(Tor(A,Z/pZ),2)(Z/pZ,Z/pZ) = H∗(K(B, 3);Z/pZ) is aconvenient way to describe this computation of Tor in a way that is familiar [2, 26].If p is odd, then H∗(K(B, 3);Z/pZ) = E(B, 3)⊗C where C is an algebra generatedby elements of degrees ≥ 3 + 2p− 2 = 2p+ 1. If p = 2 , then H∗(K(B, 3);Z/2Z) =Γ(B, 3)⊗D where D is an algebra generated by elements of degrees ≥ 5.

Hence, H2(K(A, 2), Z/pZ) = A⊗Z/pZ , H3(K(A, 2), Z/pZ) = Tor(A,Z/pZ),and, provided A⊗Z/pZ = 0, H4(K(A, 2);Z/pZ) = 0. Furthermore, if A⊗Z/pZ =Tor(A,Z/pZ) = 0, then H∗(K(A, 2), Z/pA) = Z/pZ. Thus, we have the Hurewicztheorem for all n ≥ 1 for K = K(A, 2) and k = p = a prime.

Suppose we have the mod k Hurewicz hypotheses for some n ≥ 1. By the longexact Bockstein sequence, vanishing of tensor and Tor with Z/k implies the samevanishing results for tensor and Tor with Z/dZ for all divisors d of k. Hence, themod k Hurewicz hypotheses imply the mod d Hurewicz hypotheses for all divisorsd. By induction, we can assume the truth of the mod d Hurewicz theorem forproper divisors. Now the 5-lemma applied to the long exact Bockstein sequencesimplies the mod k Hurewicz theorems for all n ≥ 1 for K = K(A, 2), that is, Hn(K)implies Cn(K) for q = 2 and for all n ≥ 1.

If q ≥ 3, the Serre spectral sequence and induction using the path fibra-tions K(A, q − 1) → PK(A, q) → K(A, q) show that Hq(K(A, q), Z/pZ) = A ⊗Z/kZ , Hq+1(K(A, q), Z/kZ) = Tor(A,Z/kZ), and, provided A ⊗ Z/kZ = 0,Hq+2(K(A, q);Z/kZ) = 0. Furthermore, if A ⊗ Z/kZ = Tor(A,Z/kZ) = 0, thenH∗(K(A, q), Z/kZ) = Z/kZ. Thus, we have the Hurewicz theorem for all n ≥ 1 forK = K(A, q) and a integer k, that is, Hn(K) implies Cn(K) for all q ≥ 1 and forall n ≥ 1.

Case 3. X = a nilpotent space with abelian fundamental group

Lemma 9.9. Let A be a nilpotent π module and set A` = IÀ/I`+1A where Iis the augmentation ideal of Z[π].

1) A⊗ Z/kZ = 0 implies A` ⊗ Z/kZ = 0 for all `.2) A ⊗ Z/kZ = Tor(A,Z/kZ) = 0 implies A` ⊗ Z/kZ = Tor(A`, Z/kZ) = 0

for all `.

Proof. Notice that IÀ is an epimorphic image of A and so is A`. Hence,A⊗ Z/kZ = 0 implies IÀ⊗ Z/kZ = 0 and A` ⊗ Z/kZ = 0 for all `.

Since IÀ is simultaneously a submodule and an epimorphic image of A, A ⊗Z/kZ = Tor(A,Z/kZ) = 0 implies IÀ⊗ Z/kZ = Tor(IÀ,Z/kZ) = 0 for all `.

Now,

0 = IÀ⊗ Z/kZ = Tor(IÀ,Z/kZ) = I`−1A⊗ Z/kZ = Tor(I`−1A,Z/kZ)

implies A` ⊗ Z/kZ = Tor(A`, Z/kZ) = 0 for all `.

Consider the Serre spectral sequence of an orientable fibration sequence K →E → B with K = K(A, q + 1), q ≥ 1. It converges to H∗(E;Z/kZ) with E2

r,s =Hr(B;Hs(K;Z/kZ)). Suppose that Hs(B;Z/kZ) = 0 for all s ≤ n − 1, thenE2s,t = Hs(B;Ht(K;Z/kZ)) = 0 for all s. Hence, there is a Serre exact sequence

Hn+q(K;Z/kZ)→ Hn+q(E;Z/kZ)→ Hn+q(B;Z/kZ)→ . . .

→ H2(K;Z/kZ)→ H2(E;Z/kZ)→ H2(B;Z/kZ)→ 0.


We have a similar exact sequence for homotopy groups with coefficients and amap from this homotopy sequence to the Serre exact sequence.

Let

X1 = K(π1(X), 1), K(A2,0, 2)→ X2,0 → X1, . . .

K(Aq,`, q)→ Xα → Xα−1, . . .

where Aq,` = I`πq(X)/i`−1πq(X) is the refinement into orientable fibrations of thePostnikov sequence of fibrations for X.

Lemma 9.10. Cn(Xα−1) and Cn(K(Aq,`, q)) imply Cn(Xα).

Proof. Assume Cn(Xα−1).For s ≤ n+ 1, consider the maps of horizontal exact sequences

πs(K(Aq,`, q);Z/kZ) → πs(Xα;Z/kZ) → πs(Xα−1;Z/kZ) →↓ φ ↓ φ ↓ φ

Hs(K(Aq,`, q);Z/kZ) → Hs(Xα;Z/kZ) → Hs(Xα−1;Z/kZ) →

If s ≤ n, the left and right mod k Hurewicz maps are isomorphisms. The mod kHurewicz map which is out of the picture to the left is an epimorphism and the oneto the right out of the picture is an monomorphism. Hence, a standard 5-lemmaargument shows that the mod k Hurewicz map in the middle is an isomorphism.

If s = 2, then we remark that our hypotheses guarantee that the homotopy setsare in fact groups and that the Hurewicz maps are homomorphisms. Otherwise,the argument is the same.

If s = n + 1, then the left and right mod k Hurewicz maps are epimorphisms.The one to the right out of the picture is an monomorphism. Hence, the stan-dard 5-lemma argument shows that the mod k Hurewicz map in the middle is anepimorphism.

The lemma is proved.

Assume Hn(X). This implies Hn(Xq) and Hn(K(Aq,`, q)) for all q ≥ 1 and all`. Hence, we have Cn(K(Aq,`, q)) for all q ≥ 1 and all `. The lemma implies that wehave Cn(Xα) for all q ≥ 1. If we let α be sufficiently large, then we have Cn(X).

In particular, the mod k Hurewicz theorem implies

Corollary 9.11. If X is a simple space (that is, the fundamental group actstrivially on all the homotopy groups) and if π1(X)⊗Z/kZ = 0, then π2(X;Z/kZ) =H2(X;Z/kZ) is an abelian group. If, in addition, π2(X;Z/kZ) = 0, then π3(X;Z/kZ) =H3(X;Z/kZ) is an abelian group.

The next section will strengthen this 3-dimensional result when k is an oddinteger.

The next result is a strengthening of the mod k Hurewicz theorem in the caseof loop spaces. Recall that π1(ΩY ;Z/kZ) = π1(ΩY )⊗ Z/kZ by definition.

Theorem 9.12 (mod k Hurewicz theorem for loop spaces). Suppose X = ΩY isa loop space and n ≥ 1. If π`(X;Z/kZ) = 0 for all ` < n, then H`(X;Z/kZ) = 0 forall ` < n and the Hurewicz map φ : πn(X;Z/kZ) → Hn(X;Z/kZ) is a bijection.If n ≥ 2, then the Hurewicz map φ : πn+1(X;Z/kZ) → Hn+1(X;Z/kZ) is asurjection.


Proof. First of all, we again recall that the case n = 1 is true almost bydefinition, π1(X;Z/kZ) = π1(X)⊗Z/kZ = H1(X)⊗Z/kZ = π1(X;Z/kZ). Hencewe begin with the case n = 2. The proof is the same as that of the mod k Hurewicztheorem for nilpotent spaces with three differences. One, all the homotopy sets areautomatically groups. Two, the loop space is simple space, that is the fundamentalgroup acts trivially on all the homotopy groups. There is no need to refine thePostnikov system to get orientable fibrations. And the Postnikov system consistsentirely of loop spaces and loop maps. These are improvements, not problems.But finally, three, the Hurewicz map φ : π2(X) → H2(X;Z/kZ) may not be ahomomorphism. But our hypotheses gurarantee that it is a homomorphism since

φ(f + g) = φ(f) = φ(g) + (βφ(f)) ∗ (βφ(g)) = φ(f) = φ(g) + (0) ∗ (0),

(βφ(f)), (βφ(g)) ∈ H1(X;Z/kZ) = 0.

Corollary 9.13. Let f : X → Y be a map of simply connected spaces. As-sume that Tor(π2(Y )/f ∗ π2(X), Z/kZ) = 0. Let n ≥ 1. Then πj(X;Z/kZ) →πj(Y ;Z/kZ) is a bijection for all j ≤ n and a surjection for j = n+ 1, if and onlyif Hj(X;Z/kZ) → Hj(Y ;Z/kZ) is a bijection for all j ≤ n and a surjection forj = n+ 1.

Remark 9.14. The example of the inclusion of the circle into the fake circleshows that we need some hypothesis to insure the truth of this result.

Proof. We can assume that f is a fibration and that F → X → Y is afibration sequence. Since the base is simply connected, the fibration is orientable.The fibration sequence ΩY → F → X → Y shows that F is nilpotent: Let I bethe augmentation ideal of the group ring Z[π1(F )]. Since X is simply connected,

Iπn(F ) ⊆ image πn(ΩY ).

Since π1(ΩY ) → π1(F ) is a surjection and ΩY is a simple space, that is, its fun-damental group acts trivially on all πn(ΩY ), F is nilpotent of length ≤ 2, that is,I2πn(F ) = 0.

Note that π1(F ) = π2(Y )/f ∗ π2(X) and hence the preliminary hypotheses ofthe mod k Hurewicz theorem are satisfied.

Assume the homotopy hypothesis. Then πj(F ;Z/kZ) = 0 for all j ≤ n. Themod k Hurewicz theorem implies that Hj(F ;Z/kZ) = 0 for all j ≤ n. The homol-ogy Serre spectral sequence shows that Hj(X;Z/kZ)→ Hj(Y ;Z/kZ) is a bijectionfor all j ≤ n and a surjection for j = n+ 1.

On the other hand, the same homology Serre spectral sequence shows that thehomology hypothesis implies that Hj(F ;Z/kZ) = 0 for all j ≤ n. The mod kHurewicz theorem implies that πj(F ;Z/kZ) = 0 for all j ≤ n.

A simpler version of the same proof shows

Corollary 9.15. Let f : X → Y be a loop map of loop spaces and suppose thatit is a monomorphism on fundamental groups. Let n ≥ 1. Then πj(X;Z/kZ) →πj(Y ;Z/kZ) is a bijection for all j ≤ n and a surjection for j = n+ 1, if and onlyif Hj(X;Z/kZ) → Hj(Y ;Z/kZ) is a bijection for all j ≤ n and a surjection forj = n+ 1.


The next result removes the requirement that the map be a loop map. Theproof is only slightly more difficult.

Corollary 9.16. Let f : X → Y be a map of loop spaces which is an iso-morphism on fundamental groups and suppose that π2(X)→ π2(Y ) is a surjection.Then πj(X;Z/kZ) → πj(Y ;Z/kZ) is an isomorphism for all j ≥ 1 if and only ifHj(X;Z/kZ)→ Hj(Y ;Z/kZ) is an isomorphism for all j ≥ 1.

Proof. Let f : X → Y be the map of universal covers. The fibration sequences

X → X → K(π1(X), 1), Y → Y → K(π1(Y ), 1)

are both orientable.Suppose that πj(X;Z/kZ) → πj(Y ;Z/kZ) is an isomorphism for all j. Since

π2(X;Z/kZ) = π2(X) ⊗ Z/kZ, π2(Y ;Z/kZ) = π2(Y ) ⊗ Z/kZ, it is easy to see

that the same equivalence is true for πj(X;Z/kZ) → πj(Y ;Z/kZ). Since thesespaces are simply connected, the previous result shows that the same equivalenceis true for Hj(X;Z/kZ)→ Hj(Y ;Z/kZ).

Since there is an isomorphism on the bases, the Zeeman comparison theorem[33, 14] applies to show that the same equivalence is true for Hj(X;Z/kZ) →Hj(Y ;Z/kZ).

The argument is reversible. Hence, we are done.

Remark 9.17. Suppose f : X → Y is a map of simply connected spaceswhich are localized at a prime p. Suppose that the homology groups H∗(X;Z) =H∗(X;Z(p)) and H∗(Y ;Z) = H∗(Y ;Z(p)) are finitely generated Z(p) modules ineach degree. Suppose also that there is some r ≥ 0 such that the induced mapf : Hj(X;Z/prZ) → Hj(Y ;Z/prZ) is an isomorphism for all j ≤ n and an epi-morphism for j = n. In other words, f is a mod pr n− equivalence. Then, if Wis any CW- complex, the map of pointed mapping sets f∗ : [W,X]∗ → [W,Y ]∗ isa bijection if the dimension of W is < n and an epimorphism if the dimension ofW is = n. The argument is: Since the homologies are of finite type, the map is ann−equivalence localized at p, therefore, it is an integral homology n−equivalence.The integral Hurewicz theorem for pairs implies that it is an integral homotopyn−equivalence. Now the classical theorem of J.H.C. Whitehead implies the result.

10. Abelian homotopy groups in dimension 3

Theorem 10.1. If k is odd, then π3(X;Z/kZ) is an abelian group.

Proof. Consider the isomorphic group π2(ΩX;Z/kZ).The commutator [ , ] : ΩX × ΩX → ΩX is given by [ω, γ] = ωγω−1γ−1.

Since [ , ] is null homotopic on the bouquet ΩX ∨ ΩX, it factors as

ΩX × ΩX → ΩX ∧ ΩX[ , ]−−−−−→ ΩX.

Let f : P 2(Z/kZ) → ΩX and g : P 2(Z/kZ) → ΩX be two maps. Thecommutator [f, g] is the composition

P 2(Z/kZ)∆−→ P 2(Z/kZ)× P 2(Z/kZ)

f×g−−−→ ΩX × ΩX[ , ]−−−−−→ ΩX


where ∆ is the diagonal. If ∆ is the reduced diagonal, this is the same as thecomposition

P 2(Z/kZ)∆−→ P 2(Z/kZ) ∧ P 2(Z/kZ)

f∧g−−→ ΩX ∧ ΩX[ . ]−−−−−→ ΩX.

Let e1 and e2 be generators of the reduced homology H∗(P2(Z/kZ);Z/kZ) of

respective dimensions 1 and 2. A computation of Steenrod [28] asserts that

∆∗(e2) =k(k + 1)

2e1 ⊗ e1

and this equals 0 when k is odd. If k is odd, then the mod k Hurewicz imageφ(∆) = 0.

The Hurewicz theorem implies that ∆ is null homotopic.Hence [f, g] is null homotopic and π2(ΩX;Z/kZ) is abelian if k is odd.

Corollary 10.2. If k is odd, then the homotopy groups with coefficients Z/kZof a loop space are all abelian groups, that is, π1(ΩX;Z/kZ) = π1(ΩX) ⊗ Z/kZand πn(ΩX;Z/kZ) = πn+1(X;Z/kZ), n ≥ 2 are all abelian groups.

LetG be a group-like space. If k is odd, the proof above shows that π2(G;Z/kZ)is an abelian group. If k is any integer, then the fact that the composition

P 2(Z/kZ)∆−→ P 2(Z/kZ) ∧ P 2(Z/kZ)

1∧ρ−−→ P 2(Z/kZ) ∧ S2

is null homotopic shows

Theorem 10.3. If k is any integer, the image of the reduction map ρ : π2(G)→π2(G;Z/kZ) is a central subgroup.

Of course, it already follows from the Hurewicz theorem that

Corollary 10.4. If k is any integer and π1(ΩX)⊗Z/kZ = 0, then the homo-topy groups with coefficients Z/kZ are all abelian groups, that is, πn(ΩX;Z/kZ)are abelian groups for all n ≥ 2.

The fake multiples allow us to prove that certain other subgroups of 3-dimensionalhomotopy groups mod 2r are central, that is

Theorem 10.5. If r ≥ 1, then the image of the 2−nd power

2∗ : π3(X;Z/2rZ)→ π3(X;Z/2rZ)

is a central subgroup.

Proof. Since the image is the same, we consider the dual of the fake multiple2∗. Images under this map are represented by compositions

P 3(Z/2rZ)2−→ P 3(Z/2rZ)

f−→ X.

Since the fake multiples are suspensions, we can take the adjoints as follows

P 2(Z/2rZ)2−→ P 2(Z/2rZ)

f−→ ΩX

where f is the adjoint of f. Hence, the images are subgroups.The desired result follows from the fact that the following maps are null homo-

topic:

P 2(Z/2rZ)∆−→ P 2(Z/2rZ)∧P 2(Z/2rZ)

2∧1−−→ P 2(Z/2rZ)∧P 2(Z/2rZ)f∧g−−→ ΩX∧ΩX

[ . ]−−−−−→ ΩX.


It is sufficient to note that the mod 2r Hurewicz image is zero. But

e2 7→2r(2r + 1)

2e1 ⊗ e1 7→

2r(2r + 1)

22e1 ⊗ e1 = 2r(2r + 1)e1 ⊗ e1 = 0

mod 2r if r ≥ 1.

11. Classical and modular Hopf invariants

In this section, we revisit the old result of Hopf [11] on the nontriviality ofπ3(S2) = Z 6= 0 and discuss a modular form of the Hopf invariant which can beused to determine whether a map P 3(Z/kZ)→ P 3(Z/`Z) is trivial or nontrivial.

The Hopf fibering η : S3 → S2 has the mapping cone S2 ∪η e4 = CP 2 and theembedding CP 2 ⊆ CP∞ yields the fibration sequence

ΩS3 Ωη−−→ ΩS2 ∂−→ S1 → S3 η−→ S2 → CP∞.

This has the immediate consequences:1) η : S3 → S2 represents a generator of the group π3(S2) ∼= Z.2) there is a homotopy equivalence

S1 × ΩS3 ι×Ωη−−−→ ΩS2 × ΩS2 mult−−−→ ΩS2.

3) if η : S2 → ΩS2 is the adjoint of η, then the Hurewicz image is φ(η) = ±x2

where H∗(ΩS2;Z) = P (x) = the polynomial algebra generated by a 2-dimensional

class x.The following are two equivalent definitions of the classical integral Hopf in-

variant H(g) of a map g : S3 → S2. (The map g : S2 → ΩS2 is the adjoint ofg.)

Definition 11.1. 1) The Hurewicz image definition:

φ(g) = H(g)φ(η).

2) The cup product definition:

u ∪ u = H(g)e

in the integral cohomology of the mapping cone Cg = S2∪ge4 where u ∈ H2(Cg;Z)and e ∈ H4(Cg;Z) are generators.

It is clear that η has Hopf invariant one in both definitions. To see that the twodefinitions are equivalent, observe that the linearity of the Hurewicz map impliesthat the first definition of H defines a linear map H1 : π3(S2)→ Z.

We claim that, for any integer `, the second definition of H is semi-linear, thatis, H2(kg) = H2(g k) = kH2(g). This implies that they are equal, H = H1 = H2.

The justification for semi-linearity is a consequence of the maps of cofibrationsequences

S3 gk−−→ S2 → Cgk → S4

↓ k ↓ 1 ↓ ↓ kS3 g−→ S2 → Cg → S4

.

Just compute the Hopf invariant via the bottom row and use naturality.

Since π3(S2) = Zη, any map g : S3 → S2 is a multiple of η, that is, g = `η.Hence,

H2(g) = H2(η `) = `H2(η) = ` = `H1(η) = H1(`η) = H1(g).


In any case, we have the classical result of Hopf [11]:

Theorem 11.2. The integral Hopf invariant is an isomorphism H : π3(S2)→Z and H(kη) = k for all integers k.

We know for trivial reasons that any suspension Σh : S3 → S2 is null homotopicbut the classical Hopf invariant gives another proof since nontrivial cup productsare zero in the cohomology ring of a suspension H∗(S2 ∪Σh e

4;Z).Inspection of the diagram

S3 g−→ S2 → Cg → S4

↓ 1 ↓ k ↓ ↓ kS3 kg−−→ S2 → Ckg → S4

shows the quadratic nature of the Hopf invariant:

Lemma 11.3. If g : S3 → S2 and k is an integer, then H(k g) = k2H(g) and,since the Hopf invariant is an isomorphism, k g ' k2g = g k2.

Our next goal is to prove a modular analog of Hopf’s theorem.

Lemma 11.4. Let A and B be finite groups. If f : Pn(A) → Pn(B) is a mapwhich induces zero in integral cohomology, that is, f∗ = 0 : B → A, then f induceszero in cohomology and homology with all coefficients.

Proof. The map f induces (f∗)∗ = 0 : A∗ → B∗ in integral homology andthe universal coefficient theorems imply that it induces zero in homology and co-homology with all coefficients.

Definition 11.5. For A and B finite, let

Kn(A,B) = f ∈ [Pn(A), Pn(B)]∗ |f∗ = 0 : B → A.

For B finite abelian, let h : Pn(B) → K(B∗, n − 1) be a map such that theintegral homology map h∗ : B∗ → B∗ is an isomorphism and let Fn(B) be thehomotopy theoretic fibre of h.

Suppose that f : Pn(A) → Pn(B) is a map. Note that h f : Pn(A) →K(B∗, n − 1) is null homotopic if and only if 0 = f∗ : A∗ → B∗. Thus, f factorsthrough the fibre Fn(B) if and only if 0 = f∗ : B → A, that is, if and only iff ∈ Kn(A,B). The fibration sequence

K(B∗, n− 2)→ F (B)→ Pn(B)→ K(B∗, n− 1)

shows that

Lemma 11.6.πn(Fn(B);A) = Kn(A,B)

We compute

Theorem 11.7. a) Let A and B be finite abelian groups with relatively primeorder, then Kn(A,B) = 0 for all n ≥ 3.

b) If p is an odd prime, then

K3(Z/prZ,Z/psZ) = Z/prZ ⊗ Z/psZ = Z/(pr, ps)Z

where (pr, ps) = the greatest common divisor of pr and ps.


c) If p is an odd prime and n ≥ 4, then

Kn(Z/prZ,Z/psZ) = 0

d)K3(Z/2rZ,Z/2sZ) = Z/2rZ ⊗ Z/2s+1Z = Z/(2r, 2s+1)Z

where (2r, 2s+1) = the greatest common divisor of 2r and 2s+1.e) If n ≥ 4, then

Kn(Z/2rZ,Z/2sZ) = Z/2Z.

Proof. a) It is easy to see that H∗(K(B∗, 1);Z) has the same exponent as theexponent of B = the exponent of B∗. Induction using the Serre spectral sequenceshows that every dimension of H∗(K(B∗, n);Z) has exponent which involves onlythe primes which divide the order of B. Since this is true of both H∗(K(B∗, n);Z)and H∗(P

n(B);Z), it is also true of the fibre H∗(F (B);Z). We use here the factthat n ≥ 3 implies that K(B∗, n − 1) is simply connected if n ≥ 3 and hence thefibration is orientable.

The Hurewicz theorem shows that π∗(Fn(B);Z/pr) = 0 for all r ≥ 1 and for

all primes p which are relatively prime to the order of B. Hence, if A is finite oforder relatively prime to the order of B, then πn(F (B);A) = 0.

b) In dimensions ∗ ≤ 5, H∗(K(Z/psZ, 2);Z/pZ) has basis

x, y = βsx, x2, xy

where x has degree 2 and with βs = the s−th cohomology Bockstein. There are noSteenrod operations yet. Since βs(x

2) = 2xy, it follows that the integral cohomol-ogy is

H∗(K(Z/psZ, 2);Z) =

Z/psZ ∗ = 3, 5

0 ∗ 6= 3, 5, ∗ ≤ 5

Hence, the integral homology is

H∗(K(Z/psZ, 2);Z) =

Z/psZ ∗ = 2, 4

0 ∗ 6= 2, 4, ∗ ≤ 4

The Serre spectral sequence yields

H∗(F3(Z/psZ);Z) =

Z/psZ ∗ = 3

0 ∗ 6= 3, ∗ ≤ 3.

It follows that

K3(Z/prZ,Z/ps) = π3(F 3(Z/psZ);Z/prZ) = Z/psZ ⊗ Z/prZ.c) The strong coefficient functoriality implies that Kn(Z/prZ,Z/psZ) = 0 if p

is an odd prime and n ≥ 4.

d) In dimensions ∗ ≤ 6, Serre’s computations [26, 19] show thatH∗(K(Z/2sZ, 2);Z/2Z)

has basisx, y = βsx, x2, Sq2y, xy, y2

where x has degree 2 and with βs = the s−th cohomology Bockstein. Recall theAdem relation Sq1Sq2y = Sq3y = y2. Since βrx

2 = 2xy = 0, Browder’s theorem[1, 10] asserts that

βr+1x2 =

xy + Sq2y mod image(Sq1) = image(β1), s = 1

xy mod image(βr), s ≥ 2


The integral cohomology is

H∗(K(Z/2sZ, 2);Z) =

Z/2sZ ∗ = 3

Z/2s+1Z ∗ = 5

0 ∗ 6= 3, 5, ∗ ≤ 5

The remainder of the argument in this case is the same as in part b).e) For n ≥ 4, Serre’s computations show that, in dimensions ∗ ≤ n + 3,

H∗(K(Z/2sZ, n− 1);Z/2Z) has basis

x, βrx = y, Sq2x, Sq2y, Sq3x, Sq3y, and, if n = 4, xy

Since Sq1Sq2 = Sq3, it follows that the integral homology is

H∗(K(Z/2sZ, n− 1);Z) =

Z/2sZ ∗ = n− 1

Z/2Z ∗ = n+ 1, n+ 2

0 ∗ 6= n− 1, n+ 1, n+ 2, ∗ ≤ n+ 2

and the rest follows as before.

Remark 11.8. The generator of Z/2Z in e) is clearly given by the composition

η : Pn(Z/2sZ)q−→ Sn

η−→ Sn−1 ι−→ Pn(Z/2sZ), n ≥ 4.

Let f : P 3(Z/kZ) → P 3(Z/`Z) be a map which induces zero in integral co-homology and therefore zero in homology and cohomology with any coefficients.That is, f ∈ K3(Z/kZ,Z/`Z). For such maps we define the modular Hopf invari-ant H(f) ∈ Z/kZ ⊗ Z/`Z = Z/(k, `)Z as follows:

Definition 11.9. Let Cf be the mapping cone of f : P 3(Z/kZ)→ P 3(Z/`Z).Let j = (k, `) = the greatest common divisor of k and ` and let u ∈ H2(Cf ;Z/jZ)and e ∈ H4(Cf ;Z/jZ) be generators. The modular Hopf invariant of f is theinteger mod j defined by the formula

u ∪ u = H(f)e.

We claim that f factors as P 3(Z/kZ)q−→ S3 g−→ S2 ι−→ P 3(Z/`Z). The vanishing

of f in integral homology shows that f is null on S2 ⊂ P 3(Z/kZ) and hence it

factors as P 3(Z/kZ)q−→ S3 → P 3(Z/`Z). Since any composition S3 → P 3(Z/`Z)→

qS3 is degree zero, it is null homotopic. Now, the Serre spectral sequence showsthat the fibre of q is approximated by S2 up to dimension 3. Hence S3 → P 3(Z/`Z)

factors as S3 → S2 P−→3

(Z/`Z).If h = g q, the commutative diagrams of cofibration sequences

P 3(Z/kZ)f−→ P 3(Z/`Z) → Cf → P 4(Z/kZ)

↓ q ↓ 1 ↓ ↓ qS3 h−→ P 3(Z/`Z) → Ch → S4

↑ 1 ↑ ι ↑ ↑ 1

S3 h−→ S2 → Cg → S4

shows that, if we choose the generator e ∈ H4(Cf ;Z/jZ)) to be compatible withthe standard generator of H4(S4;Z/jZ), then the modular Hopf invariant H(f) isthe mod j reduction of the classical Hopf invariant H(g).


Hence,

Lemma 11.10. For any integer k, H(k f) = k2H(f) and H(f k) = kH(f).

We also get that the modular Hopf invariant is always an epimorphism. Theo-rem 8.7 gives

Lemma 11.11. 1) If p is an odd prime, then the modular Hopf invariant H :K3(Z/prZ,Z/psZ)→ Z/prZ ⊗ Z/psZ is an isomorphism.

2) If p = 2, then the modular Hopf invariant H : K3(Z/2rZ,Z/2sZ) = Z/2rZ⊗Z/2s+1 → Z/2rZ ⊗ Z/2sZ is an epimorphism which is an isomorphism if r ≤ sand has kernel Z/2Z if r ≥ s+ 1.

Corollary 11.12. Let p be a prime and let f ∈ K3(Z/prZ,Z/ps). Considerthe following three statements:

1) f is null homotopic.2) f is a suspension.3) H(f) = 0 ∈ Z/prZ ⊗ Z/psZ.Then 1) =⇒ 2) =⇒ 3), and, if p is odd or if p = 2 and r ≤ s, then 3) =⇒ 1).

12. The Dold-Thom theorem

The Hurewicz map is given a geometric form as the map X → SP∞(X) whereSP∞(X) is the infinite symmetric product. Recall that SP∞(X) is the free abeliangroup generated by the points of X subject to the relation that the basepoint isthe unit.

It is filtered by the subspaces SP q(X) consisting of the words of length ≤ q.Each Spq(X) is a quotient space of the q−fold product X×· · ·×X and SP∞(X) =limSP q(X) has the direct limit topology.

If X is connected, Dold and Thom [8] proved that SP∞(X) ' K(H∗(X;Z))is a product of Eilenberg-MacLane spaces if X is connected. For n ≥ 1, the mapπn(X;Z)→ πn(SP∞(X);Z) = Hn(X;Z) is the Hurewicz map.

The proof proceeds by showing that the functor X 7→ SP∞(X) preserves ho-motopy,

f ' g =⇒ SP∞(f) ' SP∞(g),

and preserves colimits

SP∞(limXn) = limSP∞(Xn).

The key point is that the functor SP∞ converts cofibrations A → X → X/A intoquasi-fibrations

SP∞(A)→ SP∞(X)→ SP∞(X/A).

This last result is quite plausible, since if the infinite symmetric product were anabelian group, then SP∞(X/A) = SP∞(X)/Sp∞(A) would be a homogeneousspace and we would expect SP∞(A) → SP∞(X) → SP∞(X/A) to be an actualfibration.

The key feature of quasi-fibrations is that their homotopy groups satisfy longexact sequences just like those of fibrations.

Thus the cofibration sequence X → CX → ΣX gives the isomorphic connectinghomomorphism πn(SP∞(ΣX))→ πn−1(SP∞(X)).

Now it is easy to see that the functor X 7→ πn(SP∞X) satisfies the axiomsfor a reduced homology theory. If we can check that it is a connected theory with


coefficient group Z, then there must be a natural isomorphism, πn(SP∞(X)) =Hn(X;Z) for all n ≥ 1. This follows from

Lemma 12.1.SP∞(S2) ∼= CP∞.

Proof. Regard S2 = C ∪ ∞ as the Riemann sphere and define a homeomor-phism Φ : SP q(S2)→ CP q by taking the coordinates of the homogeneous complexpolynomial with roots (b1, b2, b3, . . . , bq).

That is, if the bi are all finite,

Φ(b1, b2, b3, . . . , bq) = (z + b1)(z + b2)(z + b3) . . . (z + bq) =

a0 + a1z + · · ·+ aq−1zq−1 + zq = [a0, a1, . . . , aq, 1].

If bi =∞, it is defined recursively by omitting the i−th factor,

Φ(b1, b2, b3, . . . , bq) = Φ(b1, b2, . . . , bi−1, bi+1 . . . , bq) =

(z + b1) . . . (z + bi−1)(z + bi+1) . . . (z + bq) =

a0 + a1z + · · ·+ aq−2zq−2 + zq−1 = [a0, a1, . . . , aq−2, 1, 0].

And so on until there are no more bi =∞.Thus, SP q(S2) ∼= CP q. Letting q →∞ gives the result.

SP∞(S2) = K(Z, 2) with coefficient group Z = π2SP∞(S2) and thus

πn(SP∞(X)) ∼= Hn(X;Z)

for all connected X.

Corollary 12.2. If Mn(A) is a Moore space with n ≥ 1, then SP∞(Mn(A)) 'K(A,n).

Note the obvious

Lemma 12.3. SP∞(X ∨ Y ) = SP∞(X) × SP∞(Y ) and the analogous resultfor infinitely many bouquet summands.

The following result was originally observed by John Moore. With the infinitesymmetric product, it becomes very easy.

Lemma 12.4. Let G be a strictly commutative connected topological monoidwith homotopy groups πi(G) = πi. Let

M = M(π∗) =∨i

Mi(πi, i)

be the corresponding bouquet of Moore spaces. Then there are weak homotopy equiv-alences

SP∞(M)→ G, SP∞(M)→ K(π∗(G)).

Proof. Start with a map∨Si → G, one sphere for each generator of πi.

Attaching i+ 1 cells to kill the relations gives a map Mi(πi, i)→ G and thus a map

M =∨i

Mi(πi, i)→ G.

Since G is strictly commutative, we can extend to a multiplicative map SP∞(M)→G which is a weak equivalence.

Applying this to G = K(π∗(G)) gives the second result above.


Corollary 12.5. If X is connected and M = M(H∗(X)), then there are weakhomotopy equivalences

SP∞(M)→ SP∞(X), SP∞(M)→ K(H∗(X;Z)).

For simply connected X and n ≥ 2, T. Kobayashi [13] observed: for a finitelygenerated abelian group A and simply connected X, the functor

X 7→ πn(SP∞(X);A)

satisfies the axioms for a reduced homology theory with coefficient group A =π2(SP∞(S2);A) = π2(K(Z, 2);A). Thus

πn(SP∞X;A) ∼= Hn(X;A)

for all simply connected X.Hence, for simply connected X, the map πn(X;A) → πn(SP∞(X);A) =

Hn(X;A) can be used to define the Hurewicz map with coefficients in A.This general result is satisfying but we prefer not to use it since it involves

using the fact that a connected homology theory is characterized by the Eilenberg-Steenrod axioms. We prefer to use instead the more direct and specific form of theHurewicz theorem which we have given for coefficients Z/kZ. One advantage of thelatter approach is that one does not have to restrict to simply connected spaces andto dimensions ≥ 2.

13. The fibre of the geometric Hurewicz map and uniqueness of smashdecompositions

We can use the idea of a geometric Hurewicz map to study the mod k Hurewiczmap and will refer to any map φ : X → K(H∗(X;Z)) as a geometric Hurewiczmap if

φ∗ : Hn(X;Z)∼=−→ Hn(K(Hn(X;Z), n)) ⊆ Hn(K(H∗(X;Z)))

is the natural inclusion.The case of a Peterson space is particularly simple. We embed the Peterson

space into an Eilenberg-MacLane space by attaching cells to kill all the higher

homotopy groups, Pn(Z/kZ)ι−→ K(Z/kZ, n− 1).

Lemma 13.1. If f, g : X → Pn(Z/kZ) are any maps, then a) and b) areequivalent where these are:

a) f∗ = g∗ : Hn−1(Pm(Z/kZ);Z/kZ)→ Hn−1(X;Z/kZ).

b) ι f ' ι g : X → Pn(Z/kZ)ι−→ K(Z/kZ, n− 1).

c) If, in addition, X = Pm(Z/kZ), then a) and b) are equivalent to f∗ = g∗ :Hn−1(Pm(Z/kZ);Z/kZ)→ Hn−1(Pn(Z/kZ);Z/kZ).

Proof. The equivalence of a) and b) follows from the fact that Eilenberg-MacLane spaces classify cohomology.

Suppose X = Pm(Z/kZ). Since the CW chains are

C(Pm(Z/kZ)) = < 1, βem, em >, C(Pn(Z/kZ)) = < 1, βen, en >

with differentials d(em) = kβem, d(en) = kβen, it follows that mod k cohomologyis dual to mod k homology,

Hn−1(Pm(Z/kZ);Z/kZ) ∼= Hom(Hn−1(Pm(Z/kZ);Z/kZ), Z/kZ),

Hn−1(Pn(Z/kZ);Z/kZ) ∼= Hom(Hn−1(Pn(Z/kZ);Z/kZ), Z/kZ),


and it follows that a) is equivalent to c).

The above condition on homology is often convenient but, in general, it is moreconvenient to focus on cohomology, for example

Corollary 13.2. Suppose

f, g : X →∨α

Pnα(Z/kZ)

are any maps into a finite bouquet. If f∗ = g∗ in mod k cohomology, then thecompositions below are homotopic,

ι f ' ι g : Pm(Z/kZ)→∨α

Pnα(Z/kZ)ι−→ K = ΠαK(Z/kZ, nα − 1).

In other words, the geometric Hurewicz map is a faithful representation inhomotopy of mod k cohomology.

Let p be a prime and recall the decompositions of the smash products

S1 = Pn(Z/prZ) ∧ Pm(Z/prZ) ' Pn+m−1(Z/prZ) ∨ Pn+m(Z/prZ)

valid for n,m ≥ 2 and pr 6= 2. We can also smash three of the Peterson spaces andget

S2 = Pn(Z/prZ) ∧ Pm(Z/prZ) ∧ P q(Z/prZ) '

Pn+m+q−2(Z/prZ) ∨ Pn+m+q−1(Z/prZ) ∨ Pn+m+q−1(Z/prZ) ∨ Pn+m+q(Z/prZ)

valid for n,m, q ≥ 2 and pr 6= 2.The main goal of this section is to determine exactly how much cohomology

with coefficients mod pr determines these decompositions when p is an odd prime.The result we want is the following:

Theorem 13.3. Suppose p is a prime greater than 3. Let S1 → K(H∗(S1 : Z))and S2 → K(H∗(S2 : Z)) be geometric Hurewicz maps. Let f, g : X → S1 ands, t : X → S2 be maps with f∗ = g∗ and s∗ = t∗ in mod pr cohomology. IfdimX = ` ≤ n + m = dimension of S1, then f = g + w where w is a sum ofcompositions with Whitehead products. And, if dimX = ` ≤ n+m+q = dimensionof S2, then s = t+ v where v is a sum of compositions with Whitehead products.

If p = 3, the term w is as before but, if ` = n + m + q the term v may alsoinclude a summand of order 3.

Remark 13.4. If m + n > 5, there are no nontrivial Whitehead products inthe relevant range, so that f = g above. If m = n+ q > 7, there are no nontrivialWhitehead products in the relevant range, so that s = t above if p > 3. On theother hand, if p = 3, the element of order 3 in dimension n+m+ q is a nontrivialstable element and will always be there. [20]

Before we prove this result, we will review Samelson and Whitehead products.

Definition 13.5. If f : X → G and g : Y → G are mappings into a group-like space, the (external) Samelson product is the map, unique up to homotopy,


[f, g] = [ , ] (f ∧ g) : X ∧ Y → G which factors the group commutator

X × Y f×g−−−→ G×G [ , ], (x,y)7→xyx−1y−1

−−−−−−−−−−−−−−−−−−→ G↓ ↓ ↓ 1

X ∧ Y f∧g−−→ G ∧G [ , ]−−−−−→ G

The existence and uniqueness of the Samelson product follows from the cofi-bration sequence

X ∨ Y → X × Y → X ∧ Y → ΣX ∨ ΣY → Σ(X × Y )

and the fact that ΣX ∨ ΣY → Σ(X × Y ) admits a retraction.

Definition 13.6. Let F : ΣX → Z and G : ΣX → Z be maps with respectiveadjoints f : X → ΩZ and g : X → ΩZ. The (external) Whitehead product [F,G] :ΣX ∧ Y → Z is the adjoint of the Samelson product [f, g] : X ∧ Y → ΩZ.

The following lemma is immediate:

Lemma 13.7. If G is a homotopy commutative group-like space, then anySamelson product [f, g] : X ∧ Y → G is null homotopic. If Z is an H-space,then any Whitehead product ΣX ∧ Y → Z is null homotopic.

The second half of the above lemma follows from the fact that the loops on anH-space is homotopy commutative.

Definition 13.8. For pr 6= 2, the inclusion into the smash decomposition,

∆ = ∆n,m : Pm+n(Z/prZ)→ Pm+n(Z/prZ)∨Pm+n−1(Z/prZ) ' Pn(Z/prZ)∧Pm(Z/prZ)

is called the coproduct.

Remark 13.9. If p is an odd prime, the two inclusions

∆n,m : Pm+n(Z/prZ)→ Pn(Z/prZ) ∧ Pm(Z/prZ)

andδn,m : Pm+n−1(Z/prZ)→ Pn(Z/prZ) ∧ Pm(Z/prZ)

induce in mod pr homology the maps on generators

∆n,m(em+n) = en ⊗ em, ∆n,m(βem+n) = βen ⊗ em + (−1)nen ⊗ βem,

δn,m(em+n−1) = (−1)n+1βen ⊗ em + en ⊗ βem, δn,m(βem+n) = 2βen ⊗ βem.We would like these maps to be characterized by these mod pr homology images,at least up to compositions with Whitehead products. This would follow from theunproved first theorem in this section.

This coproduct allows us to define internal Samelson products and internalWhitehead products:

Definition 13.10. If f : Pn(Z/prZ)→ G and g : Pm(Z/prZ)→ G are mapsinto a group-like space, the (internal) Samelson product [f, g] is the composition

[f, g] ∆n,m : Pm+n(Z/prZ)→ Pn(Z/prZ) ∧ Pm(Z/prZ)→ G.

Definition 13.11. If F : Pn+1(Z/prZ) → Z and G : Pm+1(Z/prZ) → Z aremaps into a space, the (internal) Whitehead product [F,G] is the composition

[F,G] Σ∆n,m : Pm+n+1(Z/prZ)→ ΣPn(Z/prZ) ∧ Pm(Z/prZ)→ Z.


Thus, the internal Samelson products and the internal Whitehead products areadjoints of one another. In both cases, we will abuse notation and use the samenotation for all these products, Samelson and Whitehead, internal and external.After all, the internal products are merely the restrictions of the external productsto bouquet summands of the domain.

Remark 13.12. Note that, if the coproduct map is characterized up to com-position with Whitehead products by its effect in mod pr homology, then theseinternal Samelson products and internal Whitehead products are well defined sinceWhitehead products vanish into an H-space.

The internal Samelson products provide a candidate for a Lie algebra structurein the homotopy groups of a group-like space G when p is a an odd prime. As yetwe have proved no Lie identities for this structure or even that it is well defined.But the Hurewicz map provides a representation φ : π`(G;Z/prZ)→ H`(G;Z/prZ)which is consistent with the Lie algebra structure in the Pontrjagin ring.

Lemma 13.13. If f : Pn(Z/prZ) → G and g : Pm(Z/prZ) → G are ho-motopy classes with internal Samelson product [f, g] : Pn+m(Z/prZ) → G, thenthe Hurewicz map is a homomorphism of Lie algebras in the sense that φ[f, g] =[φf, φg].

Proof. Since the coproduct ∆n,m : Pn+m(Z/prZ)→ Pn(Z/prZ)∧Pm(Z/prZ)is ∆∗(em+n) = en ⊗ em in mod pr homology, it suffices to check where en × em issent by the commutator map

Pn(Z/prZ)×Pm(Z/prZ)∆×∆−−−→ Pn(Z/prZ)×Pn(Z/prZ)×Pn(Z/prZ)×Pn(Z/prZ)

f×f×g×g−−−−−−→ G×G×G×G 1×T×1−−−−−→ G×G×G×G1×1×ι×ι−−−−−−→ G×G×G×G mult−−−→ G

where T is the twist map and ι is the inverse.But

en ⊗ em 7→ (en ⊗ 1 + 1⊗ en)⊗ (em ⊗ 1 + 1⊗ em) =

en ⊗ 1⊗ em ⊗ 1 + en ⊗ 1⊗ 1⊗ em + 1⊗ en ⊗ em ⊗ 1 + 1⊗ en ⊗ 1⊗ em 7→f∗en⊗1⊗g∗em⊗1+f∗en⊗1⊗1⊗g∗em+1⊗f∗en⊗g∗em⊗1+1⊗f∗en⊗1⊗g∗em 7→f∗en⊗g∗em⊗1⊗1+f∗en⊗1⊗1⊗g∗em+(−1)nm1⊗g∗em⊗f∗en⊗1+1⊗1⊗f∗en⊗g∗em 7→f∗en⊗g∗em⊗1⊗1−f∗en⊗1⊗1⊗g∗em−(−1)nm1⊗g∗em⊗f∗en⊗1+1⊗1⊗f∗en⊗g∗em 7→

(f∗en)(g∗em)− (f∗en)(g∗em)− (−1)nm(g∗em)(f∗en) + (f∗en)(g∗em) =

(f∗en)(g∗em)− (−1)nm(g∗em)(f∗en) = φ(f)φ(g)− (−1)nmφ(g)φ(f) =

[φ(f), φ(g)]

since the inverse ι(x) = −x on primitive elements x.

Let Fn → Pn(Z/prZ) → K(Z/prZ, n − 1) be the fibration sequence of thegeometric Hurewicz map with n ≥ 3. For the remainder of this section, unlessspecified otherwise, the coefficients in homology will be mod p coefficients.

With coefficients mod p, H∗(Pn(Z/prZ)) =< 1, βre, e > where the degree of e

is n and βr denotes the r−th homology Bockstein. Recall

H∗(ΩPn(Z/prZ)) = T (u, v) = UL(u, v)


where the degree of v is n− 1, the degree of u = βrv is n− 2 and UL(u, v) denotesthe universal enveloping algebra on the free Lie algebra L(u, v).

A basic result of [5] is the following:

Theorem 13.14. Let 0 → Aι−→ B

j−→ C → 0 be a short exact sequence ofgraded Lie algebras over a field in which 2 is a unit. Then the sequence of universalenveloping algebras

UA→ UB → UC

is a short exact sequence of Hopf algebras, in particular, a choice of a lift s : UC →UB defines an isomorphism of left UA modules

UA⊗ UC ι⊗s−−→ UB ⊗ UB mult−−−→ UB.

Recall that the mod p homology of Eilenberg-MacLane spaces is given by di-vided power algebras

H∗(K(Z/prZ, 1)) = Γ(u, v)

with degree of v equal to 2 and u = βrv has degree 1. (If the degree of a generator isodd, then the divided power algebra on that generator is just the exterior algebra.)If n ≥ 4,

H∗(K(Z/prZ, n− 2)) = Γ(u, v)⊗ Γ(x)⊗ . . .where the degree of v is n − 1, u = βrv has degree n − 2, and x is the element ofdegree n+2p−4 which represents the dual of the first homology Steenrod operationin the sense that P1x = u where P1 lowers degree by 2p− 2.

We note that the graded commutative algebra S(u, v) = U < u, v > is theuniversal enveloping algebra of the abelian Lie algebra < u, v > and there is amultiplicative map

S(u, v)→ Γ(u, v)

which is an isomorphism in dimensions ≤ p deg(v) if the degree of v is even andan isomorphism in dimensions ≤ p deg(βrv)) is the degree of v is odd. In the firstcase, the divided power γp(v) is not in the image and the power vp is in the kernel.In the second case, the divided power γp(u) is not in the image and the power up

is in the kernel.Let L be the kernel of the map of graded Lie algebras L(u, v) →< u, v >. It

follows from the above quoted theorem that there is an isomorphism of left ULmodules

UL⊗ S(u, v)→ T (u, v).

Since the Lie elements are all the images of Samelson products and since Eilenberg-Maclane spaces are homotopy commutative, the Lie elements all map into the fibreΩFn.

Clearly, the composition UL → H∗(ΩFn) → H∗(ΩPn(Z/prZ)) is a monomor-

phism.In the Serre spectral sequence of the loop fibration sequence

ΩFn → ΩPn(Z/prZ)→ K(Z/prZ, n− 2),

the elements u = βrv, v are infinite cycles. It is easy to see that

Theorem 13.15.H∗(ΩF3) = UL⊕ < τγp(v) >

in dimensions ≤ 2p−1 where τ is the transgression. The element τγp(v) has degree2p− 1.


This uses the fact that the elements γp(v) are primitive and must transgress.The same argument applies to the elements x below.

Theorem 13.16. If n ≥ 4

H∗(ΩFn) = UL⊕ < τx >

in dimensions ≤ n+2p−5 where τ is the transgression. The element τx has degreen+ 2p− 5.

In [5] the following two results are proved:

Theorem 13.17. Over a field in which 2 is a unit, subalgebras of free gradedLie algebras are free.

Theorem 13.18. Suppose deg(u) = deg(v)− 1. The kernel L of abelianizationL(u, v)→< u, v > is the free graded Lie algebra

L = L(adk(v)[v, u], adk(v)[u, u])k≥0

if the degree of v is even, and

L = L(adk(u)[v, v], adk(u)[u, v])k≥0

if the degree of v is odd.

When p = 3, 2p−1 = 5, and the only Lie generators in L ⊆ H∗(ΩF3) of degree≤ 5 are [u, u], [v, u], [v, [u, u]] [v, [v, u]]. Since βr is a derivation on Lie brackets andwe have anticommutativity and the Jacobi identity, we compute

βrv = u, βru = 0, βr[v, u] = [u, u],

βr[v[u, u] = [u, [u, u]] = 0,

[v, [u, u]] = [[v, u], u] + [u, [v, u]] = 2[u, [v, u]],

βr[v, [v, u]] = [u, [v, u]] + [v, [u, u]] = 3[u, [v, u]] =3

2[v, [u, u]].

Hence, no matter what the odd prime,

U < [u, u], [v, u] >→ H∗(ΩF3)

is an isomorphism in dimensions ≤ 3, < [u, u], [v, u] >= K = L mod Lie bracketsof length ≥ 3..

Recall that, if f : X → ΩZ and g : Y → ΩZ are maps with respective adjointsF : ΣX → Z and G : ΣX → Z, the multiplicative extension of the Samelsonproduct [f, g] : X ∧ Y → ΩZ is the loop map [f, g] : ΩΣX ∧ Y → ΩΣΩZ → ΩZ.As the name indicates, it extends the map on X ∧ Y. This map is the loop of thecorresponding Whitehead product Ω[F,G] : ΩΣX ∧ Y → ΩZ.

Let µ : S1 → ΩP 3(Z/prZ) and ν : P 2(Z/prZ)→ ΩP 3(Z/prZ) be the elementswhich are adjoint to the inclusions ι2 : S2 → P 3(Z/prZ) and the identity 13 :P 3(Z/prZ) → P 3(Z/prZ). The mod p Hurewicz image of the internal Samelsonproduct [ν, µ] : P 3(Z/prZ)→ ΩP 3(Z/prZ) is the Lie commutator [v, u],

φ([ν, µ]) = [ν, µ]∗e2 = [v, u].

Since the Hurewicz map commutes with the Bockstein differential, the element[u, u] is also in the image of mod p homology via

[ν, µ]∗(βre2) = βr[ν, µ]∗e2 = βr[v, u] = [u, u].


The homology map of this Peterson space has an image of rank 2. The multiplicativeextension [ν, µ] : ΩΣP 3(Z/prZ)→ ΩP 3(Z/prZ) gives the homology map

H∗(ΩΣP 3(Z/prZ)) = UL([u, u], [v, u])→ H∗(ΩF3).

The internal Whitehead product [13, ι2] : P 4(Z/prZ) → P 3(Z/prZ) vanisheswhen projected to the Eilenberg-MacLane space and thus factors through the fibreF3. Now it loops to give the map ΩP 4(Z/prZ) → ΩF3 which is the same map asthe multiplicative extension of the internal Samelson product and which in mod phomology is the map

H∗(ΩP4(Z/prZ)) = UL([u, u], [v, u])→ H∗(ΩF3).

This is a mod p homology isomorphism in dimensions ≤ 3. Since these spaces arelocalized at p, it is a weak equivalence in the sense that it is a homotopy isomorphismin dimensions ≤ 2 and a homotopy epimorphism in dimension 3. Hence,

Lemma 13.19. a) If p is an odd prime, the Whitehead product [13, ι2] factorsas

P 4(Z/prZ)→ F3 → P 3(Z/prZ)

where the first map is a homotopy isomorphism in degrees ≤ 3 and a homotopyepimorphism in degree 4.

Corollary 13.20. If p is an odd prime, then any multiple ps : P 3(Z/prZ)→P 3(Z/prZ) is homotopic to the fake multiple ps.

Corollary 13.21. If p is an odd prime and s ≤ r, then the composition

P 3(Z/prZ)η−→ P 3(Z/psZ)

ρ−→ P 3(Z/prZ)

is homotopic to the multiple pr−s plus a possible sum of compositions with White-head products. Similarly, the composition

P 3(Z/psZ)ρ−→ P 3(Z/psZ)

η−→ P 3(Z/prZ)

is homotopic to the multiple pr−s plus a possible sum of compositions with White-head products.

Remark 13.22. If the dimension is ≥ 4 in the two corollaries above, thencoefficient functoriality implies that the indicated maps are homotopic without theneed of sums of compositions with Whitehead products.

Let

P =

P 2n−1(Z/prZ) if n is even

P 2n−2(Z/prZ) if n is odd.

Note that P is the domain of the Whitehead product [1n, 1n] if n is even, and it is thedomain of the Whitehead product [1n, ιn−1] if n is odd, where 1n : Pn(Z/prZ) →Pn(Z/prZ) is the identity and ιn−1 : Sn−1 → Pn(Z/prZ) is the inclusion.

Theorem 13.23. If p ≥ 5 and n ≥ 4, the respective Whitehead product factorsas P → Fn → Pn(Z/prZ) where the P → Fn is an n + 3 equivalence, that is,a homotopy isomorphism in dimensions ≤ n + 4 and a homotopy epimorphism indimension n+ 5.

Remark 13.24. If p ≥ 5 and n ≥ 9, then P is n + 5 connected and hence∗ → Fn is an n+ 5 equivalence.


The situation with p = 3 is slightly more complicated. Denote by α1 the firstnonzero 3-primary homotopy class α1 : Sn+2 → Sn−1 ⊂ Pn(Z/3rZ). It has order3.

Theorem 13.25. If p = 3 and n ≥ 4, the respective Whitehead product factorsas P → Fn → Pn(Z/3rZ) and α1 factors as Sn+2 → Fn → Pn(Z/3rZ) which givesan n+ 2 equivalence P ∨ Sn+2 → Fn.

Remark 13.26. Just as before, if p = 3 and n ≥ 6, we may omit P and thereis an n+ 2 equivalence Sn+2 → Fn.

Proof. Assume p ≥ 5 and n ≥ 4.Since

H(ΩFn) = UL⊕ < τx >

in dimensions ≤ n+2p−5, H(ΩFn) = UL in dimensions ≤ n+2p−6. The minimumvalue of 2p− 6 is 4.

When K = L modulo products of length ≤ 3, H(ΩFn) = UK in dimensions≤ min(3n − 6, n + 4). The minimum value is n + 2. Hence, H(ΩFn) = UK indimensions ≤ n+ 2.

Thus, ΩP → ΩFn is an n+ 2 equivalence and P → Fn is an n+ 3 equivalence.Assume p = 3 and n ≥ 4. Then

H(ΩFn) = UL⊕ < τx >= UK⊕ < τx >

in dimensions ≤ n+ 1.Recall that the transgression τx detects α1

Sn+1 → ΩSn−1, n even

Sn+1 → Sn−2, n odd

via one of the fibration sequencesF → ΩSn−1 → K(Z, n− 2), n even

F → Sn−2 → K(Z, n− 2), n odd

It follows from naturality that τx is detecting the factorization of α1 : Sn+1 →ΩFn → ΩPn(Z/3rZ).

Hence, Ω(P ∨ Sn+2)→ ΩFn is an n+ 1 equivalence and P ∨ Sn+2 → Fn is ann+ 2 equivalence.

We are now in a position to use the Hilton-Milnor theorem to prove the mainresult of this section.

RecallS1 = Pn+m−1(Z/prZ) ∨ Pn+m(Z/prZ) = P1 ∨ P2

valid for n,m ≥ 2 and the geometric Hurewicz map

S1 → K = K1 ×K2 = K(Z/prZ, n+m− 1)×K(Z/prZ, n+m− 2).

Recall also

S2 = Pn+m+q−2(Z/prZ)∨Pn+m+q−1(Z/prZ)∨Pn+m+q−1(Z/prZ)∨Pn+m+q(Z/prZ) =

P1 ∨ P2 ∨ P3 ∨ P4

and the geometric Hurewicz map

S2 → K1 ×K2 ×K3 ×K4 =


K(Z/prZ, n+m+ q − 1)×K(Z/prZ, n+m+ q − 2)×K(Z/prZ, n+m+ q − 2)×K(Z/prZ, n+m+ q − 3).

In all the above cases, let Fi → Pi → Ki denote the fibration sequence.

Theorem 13.27. Suppose p is a prime greater than 3. Let h : X → S1 andj : X → S2 be maps with h∗ = 0 and j∗ = 0 in mod pr cohomology. If dimX =` ≤ m+ n = dimension of S1, then h = w where w is a sum of compositions withWhitehead products. And, if dimX = ` ≤ n+m+ q = dimension of S2, then j = vwhere v is a sum of compositions with Whitehead products.

If p = 3, the term w is as before but, if ` = n + m + q the term v may alsoinclude a summand of order 3.

Proof. One form of the Hilton-Milnor theorem asserts that there is a fibrationsequence ∨

k≥0

ΣY ∧k ∧ Z → Σ(Y ∨ Z)→ ΣY

where the fibre is mapped in by a bouquet of the Whitehead products adk(1ΣY )(1ΣZ).Repeating this gives a fibration sequence∨

`≥0,k>0

ΣZ∧` ∧ Y ∧k ∧ Z → Σ(Y ∨ Z)→ ΣY × ΣZ

where the fibre is mapped in by a bouquet of Whitehead products.Suppose that h : X → S1 is such that h∗ = 0 in mod pr cohomology.Consider the composition

Xh−→ P1 ∨ P2

ι1∨ι2−−−→ P1 × P2j1×j2−−−−→ K1 ×K2.

Since h∗ = 0, it follows that the composition

(j1 × j2) (ι1 ∨ ι2) h ' 0

and hence (ι1 ∨ ι2) h factors through the fibre F1 × F2. Thus,

(ι1 ∨ ι2) h = (y1, y2)

where y1 and y2 are both sums of compositions with Whitehead products.Hence,

(ι1 ∨ ι2) h− ι1 y1 − ι2 y2 = 0

and h − ι1 y1 − ι2 y2 factors through the fibre of P1 ∨ P2 → P1 × P2. By theHilton-Milnor theorem,

h− ι1 y1 − ι2 y2 = z

where z = a sum of compositions with Whitehead products, that is, h = ι1 y1 +ι2 y2 + z is a a sum of compositions with Whitehead products.

The case of j∗ = 0 is similar. The iteration of the Hilton-Milnor theorembecomes more complicated but the fibre of the 4-fold bouquet into the 4-fold productis still a bouquet of Whitehead products.

The only extra fact required is that, if p = 3 and dimX = ` = m+ n+ q, thenany map Pm+n+q(Z/3rZ)→ Sm+n+q is a suspension and thus composition with

α1 : Sm+n+q → Sm+n+q−3 ⊆ Pm+n+q2(Z/3rZ)

has order 3.


14. Samelson products and Lie identities in groups

Let G be a group-like space and let X, Y, and Z be pointed spaces. Thehomotopy sets [X,G] are groups and there are certain natural subgroups related tothe definition of Samelson products, that is,

Lemma 14.1. The natural map j∗ : [X∧Y,G]∗ → [X×Y,G] is a monomorphismand the domain can be identified via this monomorphism with the subgroup K ⊆[X × Y,G]∗ where

K = ker [X × Y,G]→ [X ∨ Y,G]∗

is the kernel of restriction to the wedge.Similarly, the natural map j∗ : [X∧Y ∧Z,G]∗ → [X×Y ×Z,G] is a monomor-

phism and the domain can be identified via this monomorphism with the subgroupK ′ ⊆ [X × Y×, G]∗ where

K ′ = ker [X × Y × Z,G]→ [(X × Y × ∗) ∪ (X × ∗ × Z) ∪ (∗ × Y × Z), G]∗

is the kernel of restriction to the fat wedge.

Proof. The first statement follows from the cofibration sequence

X ∨ Y → X × Y → X ∧ Y → ΣX ∨ ΣY → Σ(X × Y )

and the fact that ΣX ∨ ΣY is a retract of

Σ(X × Y ) ' ΣX ∨ ΣY ∨ Σ(X ∧ Y ).

The second statement follows from the cofibration sequence

(X × Y × ∗) ∪ (X × ∗ × Z) ∪ (∗ × Y × Z)→ X × Y × Z → X ∧ Y ∧ Z →Σ((X × Y × ∗) ∪ (X × ∗ × Z) ∪ (∗ × Y × Z))→ Σ(X × Y × Z)

and the fact that

Σ((X × Y × ∗) ∪ (X × ∗ × Z) ∪ (∗ × Y × Z))

is a retract of Σ(X × Y × Z).

Definition 14.2. If f : X → G and g : Y → G are two maps, their Samelsonproduct

[X,G]∗ × [Y,G]∗ → [X ∧ Y,G]∗ ⊆ [X × Y,G]∗, (f, g) 7→ [f, g]

has the following two equivalent descriptions in terms of the commutator c(f, g) =[f, g] : c(f, g)(x, y) = [f(x), g(y)] = f(x)g(y)f(x)−1g(y)−1, (x, y) ∈ X × Y.

a) [f, g] : X ∧ Y → G is the element, unique up to homotopy, such that

c(f, g) = [f, g] j : X × Y → X ∧ Y → G.

b) c(f, g) = [f, g] : X × Y → G regarded as lying in the subgroup ker [X ×Y,G]∗ → [X ∨ Y,G]∗.

Note that the above is a commutator of length 2 and in two variables x, y.

Similarly, if f : X → G, g : Y → G, and h : Z → G are three maps, theirSamelson product

[X,G]∗×[Y,G]∗×[Z.G]∗ → [X∧Y ∧Z,G]∗ ⊆ [X×Y ×Z,G]∗, (f, g, h) 7→ [f, [g, h]]


has the following two equivalent descriptions in terms of the commutator c′(f, g, h) =[f, [g, h]] : c′(f, g, h)(x, y, z) = [f(x), [g(y), h(z)]], (x, y, z) ∈ X × Y × Z.

a) [f, [g, h]] : X ∧ Y ∧Z → G is the element, unique up to homotopy, such that

c′(f, g, h) = [f, [g, h]] j : X × Y × Z → X ∧ Y ∧ Z → G.

b) c′(f, g, h) = [f, [g, h]] : X × Y × Z → G regarded as lying in the subgroup

ker [X × Y × Z,G]→ [(X × Y × ∗) ∪ (X × ∗ × Z) ∪ (∗ × Y × Z), G]∗

This is a commutator of length 3 in three variables x, y, z.

Remark 14.3. The Samelson products [f, g] : X∧Y → G are sometimes calledexternel Samelson products in order to distinguish them from the internal Samelsonproducts in the next section.

Definition 14.4. A pointed space X is coabelian if the reduced diagonal map

∆ : X∆−→ X ×X j−→ X ∧X is null homotopic.

Remark 14.5. The reduced diagonal should not be confused with the comul-tiplication ∆ = ∆n,m : Pn+m(Z/prZ)→ Pn(Z/prZ) ∧ Pm(Z/prZ).

Remark 14.6. Examples of coabelian spaces are: suspensions, co-H-spaces,P 2(Z/kZ) with k = pr 6= 2, a power of a prime. The last example is not a co-H-space since:

A factorization X → X ∨X → X ×X of the diagonal implies a factorizationof the fundamental groups. In the above case, this would be a factorization

Z/kZ → (Z/kZ) ∗ (Z/kZ)→ (Z/kZ)× (Z/kZ).

Since the only elements of finite order in the free product lie in conjugates of thetwo Z/kZ factors [15], this is impossible.

The next result is a generalization of a result on homotopy groups with coeffi-cients, π2(ΩW ;Z/kZ) is an abelian group if k = pr 6= 2.

Theorem 14.7. G grouplike, X coabelian implies [X,G]∗ is an abelian group.

Proof. If f ;X → G and g : X → G are maps, then the commutator

[f, g] : X∆−→ X ∧X f∧g−−→ G ∧G [ , ]−−−−−→ G, x 7→ [f(x), g(x)]

is null homotopic.

It is worthwhile reflecting on the above proof. We showed that a length 2commutator [f(x), g(x)] in one variable vanishes in the group [X,G]∗ when X iscoabelian. We call a commutator c(f1, f2, . . . , fn) a simple commutator if each ofthe fi = fi(xi) is a function of one variable. We call it a commutator in k variablesif the number of distinct xi is k. We allow the possibility that a commutator mayhave some inverses in it. For example, [a, b]−1, [a−1, b], [a, [b−1, c]], [[a, b]−1, [c, d]]are commutators of respective lengths 2,2,3, and 4.

The same proof as above shows that

Theorem 14.8. If X1, X2, . . . , Xn are coabelian spaces, then a simple com-mutator of length > n in n variables vanishes in the group [X1∧X2∧ · · ·∧Xn, G]∗.


Proof. The simple commutator of length k > n is defined by a composition

X1 ∧X2 ∧ · · · ∧Xn → Y1 ∧ Y2 ∧ · · · ∧ Ykf1∧f2∧···∧fk−−−−−−−−→ G ∧G ∧ · · · ∧G c−→ G.

The Yi are chosen from among the Xi. Since k > n, there must be at least onerepetition of the Xi in the list of Yi and thus at least one occurence of the reduceddiagonal in the map

X1 ∧X2 ∧ · · · ∧Xn → Y1 ∧ Y2 ∧ · · · ∧ Yk.Hence, it and all compositions with it are null homotopic.

For example, [f(x), [g(x), h(y)] is null homotopic in the group [X ∧Y,G]∗ sincethe length 3 is greater than the number of variables 2. Similarly, [f(x), [g(y), [h(y), k(z)]]]and [[f(x), g(y)]−1, [h(y), k(z)]] are null homotopic in the group [X ∧ Y ∧ Z,G]∗.

Serre’s book [27] contains a list of the identities of Lie type for the commutatorsin a group.

Let a, b, c be elements of a group G and define

1) The congugate homomorphisms are ab = b−1ab. Recall that (ab)c = acbc

and (ab)b−1

= a.

2) The commutators are [a, b] = aba−1b−1. Thus, [a, b]c = [ac, bc].

The Lie identities in groups are the following formulas:

Lemma 14.9. For elements a, b, c in a group G,1) exponentiation modulo a commutator:

ab = a[a−1, b−1]

2) inverse of a commutator:

[a, b]−1 = [b, a], [a−1, b] = [b, a]a

3) commutativity modulo commutators:

ab = [a, b]ba

4) bilinearity modulo commutators:

[a, bc] = [a, b] [a, c](b−1), [ab, c] = [b, c](a

−1) [a, c]

5) Jacobi identity modulo commutators

[a(b−1), [c, b]] [b(c−1), [a, c]] [c(a

−1), [b, a]] = 1

Remark 14.10. It is difficult to discover some of the above formulas and it isdifficult to remember their exact form, but there can be no doubt that they arestraightforward to prove. Apply the procedure of reducing a word in a free groupto the identity. That is, write them in the form c = 1 and reduce the word c to theidentity via successive applications of

1) wdw−1 = 1 if and only if d = 1.2) ww−1 = 1.

Let G be a group-like space and let X,Y, Z be coabelian spaces. The multi-plication in G, (g, h) 7→ gh induces an abelian operation in each [X,G]∗, writtenadditively as (f, g) 7→ x+ y ≡ xy. Let

f, f1 : X → G, g, g1 : Y → G, h : Z → G


be maps. The next result is a variation of a result of George Whitehead [30]. Thefact that we have domains which are coabelian transforms the above Lie identi-ties for groups into the standard (ungraded) Lie algebra identities for Samelsonproducts.

Theorem 14.11. Each [X,G]∗ is an abelian group with x+ y ≡ xy, −x ≡ x−1

and 0 ≡ 1.1)

[g, f ] = −[f, g] Tin [Y ∧X,G]∗ where T : Y ∧X → X ∧ Y, y ∧ x 7→ x ∧ y is the twist.

2)[f + f1, g] = [f, g] + [f1, g], [f, g + g1] = [f, g] + [f, g1]

in [X ∧ Y,G]∗.3)

[f, [h, g]] + [g, [f, h]] σ + [h, [g, f ]] σ2 = 0

in [X ∧ Z ∧ Y,G] where

σ : X ∧ Z ∧ Y → Y ∧X ∧ Z, x ∧ z ∧ y 7→ y ∧ x ∧ zis the cyclic permutation.

Proof. 1) Since [f(x), g(y)]−1 ' [g(y), f(x)], the maps [f, g] T : Y ∧ X →X ∧ Y → G and [g, f ] : Y ∧X → G are homotopic.

2) Since

[f(x), g(y)g1(y)] = [f(x), g(y)][f(x), g1(y)]g(y)−1

= [f(x), g(y)][f(x), g1(y)][[f(x), g(y)]−1, g(y)]

and[[f(x), g(y)]−1, g(y)] = 0

since it is a simple length 3 commutator in 2 variables,

[f, g + g1] = [f, gg1] = [f, g][f, g1] = [f, g] + [f, g1]

in [X ∧ Y,G]∗.By 1), this is sufficient, that is,

[f + f1, g] = [g, f + f1] T = ([g, f ] + [g, f1]) T =

[g, f ] T + [g, f1] T = [f, g] + [f1, g].

3) We know that

[f(x)(g(y)−1), [h(z), g(y)]] [g(y)(h(z)−1), [f(x), h(z)]] [h(z)(f(x)−1), [g(y), f(x)]] = 1.

We claim that

[f(x)(g(y)−1), [h(z), g(y)]] = [f(x), [g(y), h(z)]]

and, symmetrically,

[g(y)(h(z)−1), [f(x), h(z)]] = [g(y), [f(x), h(z)]], [h(z)(f(x)−1), [g(y), f(x)]] = [h(z), [g(y), f(x)]].

Then the first line translates to

[f, [h, g]] + [g, [f, h]] σ + [h, [g, f ]] σ2 = 0.

in [X ∧ Z ∧ Y,G]∗But

[f(x)(g(y)−1), [h(z), g(y)]] = [f(x)[f(x)−1, g(y)−1], [h(z), g(y)]] =


[[f(x)−1, g(y)−1], [h(z), g(y)]]f(x)−1

[f(x), [h(z), g(y)]] = [f(x), [h(z), g(y)]].

We use that [[f(x)−1, g(y)−1], [h(z), g(y)]] and all its conjugates are zero sinceit is a simple commutator of length 4 in 3 variables.

15. Internal Samelson products

Let G be a group-like space and p an odd prime.In this section, we are going to discuss internal Samelson products

[ , ] : πn(G;Z/prZ)⊗ πm(G;Z/prZ)→ πn+m(G;Z/prZ)

[ , ] : πn(G;Z)⊗ πm(G;Z/prZ)→ πn+m(G;Z/prZ)

[ , ] : πn(G;Z/prZ)⊗ πm(G;Z)→ πn+m(G;Z/prZ)

[ , ] : πn(G)⊗ πm(G)→ πn+m(G).

These constructions are based on external Samelson products. In order to avoidconfusion, in this section external Samelson products will be distinguished with asubscript, that is,

[f, g]e : Pn(Z/prZ) ∧ Pm(Z/prZ)→ G

when f : Pn(Z/prZ) → G and f : Pm(Z/prZ) → G. Similarly, we have externalSamelson products

[f, g]e : Sn ∧ Pm(Z/prZ)→ G

[f, g]e : Pn(Z/prZ) ∧ Sm → G

[f, g]e : Sn ∧ Sm → G

The passage from external Samelson products to internal Samelson products isvia comultiplications

∆n,m : Pn+m(Z/prZ)→ Pn(Z/prZ) ∧ Pm(Z/prZ)

∆n,m : Pn+m(Z/prZ)→ Sn ∧ Pm(Z/prZ)

∆n,m : Pn+m(Z/prZ)→ Pn(Z/prZ) ∧ Sm

∆n,m : Sn+m → Sn ∧ Sm

The last three of these coproducts are just the standard homeomorphisms. All fourare characterized by the effect in homology:

∆∗(em+n) = en ⊗ em.Even in the lowest dimensions, homology characterizes these maps up to homotopyand up to composition with Whitehead products. Thus, the internal product [f, g]which arises by composition, for example,

[f, g] = [f, g]e ∆ : Pn+m(Z/prZ)∆−→ Pn(Z/prZ) ∧ Pm(Z/prZ)

[f,g]e−−−→ G

has codomain in an H−space and is well defined.We have already checked that the Hurewicz map φ : πn(G;Z/prZ)→ Hn(G;Z/prZ)

is a map of graded Lie algebras, that is, the Hurewicz map is an additive homo-morphism (p is odd!) and it respects the Lie products

φ([f, g]) = [φf, φg] = (φf)(φg)− (−1)deg(f)deg(g)(φg)(φf).

When 2 is a unit in the ground ring, the definition of a graded Lie algebra is


Definition 15.1. Let R be a commutative ring in which 2 is a unit. A gradedLie algebra L = L∗ over R is a graded R−module together with bilinear pairings

[ , ] : Lm ⊗ Ln → Lm+n

such that1)

[x, y] = −(−1)deg(x)deg(y)[y, x].

2) if the degree of x is even, the double vanishing identity is valid:

[x, x] = 0.

3) if the degree of x is odd, the triple vanishing identity is valid:

[x, [x, x]] = 0.

4)

[x, [y, z]] = [[x, y], z] + (−1)deg(x)deg(y)[y, [x, z]].

Remark 15.2. If 2 is a unit in the ring, it is clear that 1) implies 2) since:

[x, x] = −[x, x]

when x has even degree.If 3 is a unit in the ring, then 1) and 4) imply 3) since:

[x, [x, x]] = [[x, x], x]− [x, [x, x]] = −2[x[x, x]]

when x has odd degree.

Let f, f1 be chosen from either πn(G;Z/prZ), n ≥ 2, or from πn(G), n ≥ 1.Similarly, let g, g1 be chosen from either πm(G;Z/prZ),m ≥ 2, or from πm(G),m ≥1, and let h be chosen from either πq(G;Z/prZ), q ≥ 2, or from πq(G), q ≥ 1. Then

Theorem 15.3. a) Internal Samelson products are bilinear:

[f + f1, g] = [f, g] + [f1, g], [f, g + g1] = [f, g] + [f, g1].

b) Samelson products are anti-commutative:

[f, g] = −(−1)nm[g, f ].

c) If p 6= 3 the Jacobi identity is satisfied:

[f, [g, h]] = [[f, g], h] + (−1)nm[g, [f, h]].

If p = 3, the Jacobi identity is valid up to an error of order 3.

Proof. a) Since

[f + f1, g]e = [f, g]e + [f1, g]e

for external products and since the addition is defined by the multiplication in G,the internal products satisfy

[f + f1, g] = [f + f1, g]e ∆ = [f, g]e ∆ + [f1, g]e ∆ = [f, g] + [f1, g].

b) The diagram below commutes in mod pr homology and therefore it

Pn+m(Z/prZ)∆−→ Pn(Z/prZ) ∧ Pm(Z/prZ)

↓ (−1)nm ↓ TPn+m(Z/prZ)

∆−→ Pm(Z/prZ) ∧ Pn(Z/prZ)


commutes up to composition with Whitehead products. The diagrams which cor-respond to the other choices of coefficients also commute.

Hence,

[f, g] = [f, g]e ∆ = −[g, f ]e T ∆ = −(−1)nm[g, f ]e ∆ = −(−1)nm[g, f ].

c) Write ∆ = (1 ∧ ∆) ∆. If p 6= 3, the diagram below commutes in mod pr

homology and therefore it commutes up to composition with Whitehead products.(Here Pn = Pn(Z/prZ)):

Pn+m+q ∆−→ Pn ∧ Pm+q 1∧∆−−−→ Pn ∧ P q ∧ Pm↓ 1 ↓ 1

Pn+m+q ∆−→ Pn+q ∧ Pm ∆∧1−−−→ Pn ∧ P q ∧ Pm↓ 1 ↓ T ↓ σ

Pn+m+q ∆−→ Pm ∧ Pn+q 1∧∆−−−→ Pm ∧ Pn ∧ P q.

Thus, σ ∆ = (−1)m(n+q)∆. And

σ2 ∆ = (−1)m(n+q)σ ∆ = (−1)m(n+q)+q(m+n)∆ = (−1)(m+q)n∆.

After we apply right composition with ∆ to the Jacobi identity for externalSamelson products, we get the equation

[f, [h, g]e]e ∆ + [g, [f, h]e]e σ ∆ + [h, [g, f ]e]e σ2 ∆ = 0.

This is equivalent to

[f, [h, g]e]e ∆ + (−1)n+m)q[g, [f, h]e]e ∆ + (−1)(m+q)n[h, [g, f ]e]e ∆ = 0.

Since [f, [h, g]] = [f, [h, g]e]e ∆, we get the equation of internal products

[f, [h, g]] + (−1)n+m)q[g, [f, h]] + (−1)(m+q)n[h, [g, f ]] = 0.

But, in the presence of 2) above, this equation is equivalent to

[f, [h, g]] = [[f, h], g] + (−1)nq[h, [f, g]].

When p = 3, there is an error of order 3 in the initial equations for compositionsσ ∆ and σ2 ∆. (See the last theorem in Section 13.) This leads to an error oforder 3 in the Jacobi identity.

Remark 15.4. Since the diagram below commutes up to composition withWhitehead products

Pn+m(Z/prZ)ρ−→ Pn+m(Z/prZ)

ρ−→ Sn+m

↓ ∆ ↓ ∆ ↓ ∆

Pn(Z/prZ) ∧ Pm(Z/prZ)1∧ρ−−→ Pn(Z/prZ) ∧ Sm ρ∧1−−→ Sn ∧ Sm

it follows that, for f : Sn → G, g : Sm → G, and h : Pn(Z/prZ)→ G,

ρ[f, g] = [ρf, g], ρ[h, g] = [h, ρg].


In the diagram below, consider the three horizontal compositions

−→ ∆−→∼=−→ −→ Sn−1 ∧ Pm(Z/prZ)

∼= ↓ β ∧ 1

Pn+m−1(Z/P rZ)ρ−→ Sn−1 β−→ Pn+m(Z/prZ)

∆−→ Pn(Z/prZ) ∧ Pm(Z/prZ)∼= ↑ 1 ∧ β

−→ ∆−→∼=−→ −→ Pn(Z/prZ) ∧ Sm−1.

The sum β ∧ 1 + (−1)n1∧β induces the same map in mod pr homology as the map∆ β ρ. Hence, they are equal up to composition with Whitehead products. Thisyields the derivation formula for the Bockstein

Theorem 15.5. If f : Pn(Z/prZ) → G and g : Pm(Z/prZ) → G are maps,then

β[f, g] = [βf, g] + (−1)n[f, βg].

Definition 15.6. Let 2 be a unit in the ground ring. A differential graded Liealgebra is a graded Lie algebra L together with a degree -1 linear map d : Lm →Lm−1 which is a differential, d2 = 0, and a derivation,

d[x, y] = [dx, y] + (−1)deg(x)[x, dy].

If we set π1(G;Z/prZ) = π1(G) ⊗ Z/prZ, and adopt the convention that theBockstein on a 1 dimensional class is zero, then we get

Theorem 15.7. If p is a prime greater than 3 and G is a group-like space, thenthe composition d = ρ β : πm(G;Z/prZ) → πm−1(G) → πm−1(G;Z/prZ) makesπ∗(G;Z/prZ), ∗ ≥ 1 into a differential graded Lie algebra over the ring Z/prZ.

Remark 15.8. If f : S1 → G and g : S1 → G are one-dimensional classes,[f, g] is defined mod pr as a composition

P 2(Z/prZ)ρ−→ S1 ∧ S1 h=[f,h]−−−−−→ G.

If g : Pn(Z/prZ)→ G, then [f, g] is defined mod pr as a composition

Pn+1(Z/prZ)ρ−→ S1 ∧ Pn(Z/prZ)

h=[f.g]−−−−−→ G.

In order to justify the above theorem, we need that, if f is a one-dimensional class,then [prf, g] = 0 mod pr. In either case,

[prf, g] = h (pr ∧ 1) q = h 0 = 0.

Remark 15.9. If p = 3, π∗(G;Z/3rZ) is a differential graded Lie algebraexcept that the Jacobi identity is valid only up to an error of order 3. But theJacobi identity is valid if any of the three elements in the identity is the reductionof an integral class, that is, if any of the classes factors as

Pn(Z/3rZ)→ Sn → G

or, if n = 1, is represented by S1 → G.If p = 3, the triple vanishing identity [x, [x, x]] = 0 for an odd class x is valid

up to an error of order 9. The Jacobi identity implies that, if X is an odd class,then 3[x, [x, x]] = 0. When p = 3, the fact that the Jacobi identity contains anerror of order 3 implies that 9[x, [x, x]] = 0.


If p is an odd prime and s ≤ r, the diagram below commutes up to compositionswith Whitehead products since it commutes in mod pr cohomology

Pn+m(Z/psZ)∆−→ Pn(Z/psZ) ∧ Pm(Z/psZ)

↓ ρ ↓ ρ ∧ ρPn+m(Z/prZ)

∆−→ Pn(Z/prZ) ∧ Pm(Z/prZ)

In fact, they commute on the chain level:

en+m∆∗−−→ en ⊗ em

↓ ρ∗ ↓ ρ∗ ⊗ ρ∗en+m

∆∗−−→ en ⊗ em

en+m−1∆∗−−→ en−1 ⊗ em + (−1)nen ⊗ em−1

↓ ρ∗ ↓ ρ∗ ⊗ ρ∗pr−sen+m−1

∆∗−−→ pr−sen−1 ⊗ em + (−1)nen ⊗ pr−sem−1

We can replace a Peterson space by a sphere in the above, that is, if s ≤ r, thefollowing commutes up to compositions with Whitehead products

Pn+m(Z/psZ)∆−→ Pn(Z/psZ) ∧ Pm(Z/psZ)

↓ ρ ↓ ρ ∧ ρPn+m(Z/prZ)

∆=1−−−→ Sn ∧ Pm(Z/prZ)

Similarly, if p is an odd prime and s ≤ r, the diagram below commutes up tocomposition with Whitehead products


↓ η ↓ η ∧ η

Pn+m(Z/psZ)∆pr−s−−−−−→ Pn(Z/psZ) ∧ Pm(Z/psZ)

Hence,

Theorem 15.10. If p is an odd prime, s ≤ r and G is a group-like space, thefollowing diagrams commute

πn(G;Z/prZ)⊗ πm(G;Z/prZ)[ , ]−−−−−→ πm+n(G;Z/prZ)

↓ ρ⊗ ρ ↓ ρπn(G;Z/psZ)⊗ πm(G;Z/psZ)

[ , ]−−−−−→ πm+n(G;Z/psZ)

πn(G)⊗ πm(G;Z/prZ)[ , ]−−−−−→ πm+n(G;Z/prZ)

↓ ρ⊗ ρ ↓ ρπn(G;Z/psZ)⊗ πm(G;Z/psZ)

[ , ]−−−−−→ πm+n(G;Z/psZ)

πn(G;Z/psZ)⊗ πm(G;Z/psZ)[ , ]−−−−−→ πm+n(G;Z/psZ)

↓ η ⊗ η ↓ pr−sηπn(G;Z/prZ)⊗ πm(G;Z/prZ)

[ , ]−−−−−→ πm+n(G;Z/prZ)


16. Universal models and relative Samelson products

If G is a group-like space, Samelson products πn(G;Z/prZ)⊗πm(G;Z/prZ)→πn+m(G;Z/prZ) are topological analogs of the commutator maps [ , ] : G×G→G, (x, y) 7→ xyx−1y−1 for groups.

If H ⊆ G is a normal subgroup, then the commutator compresses, that is,[ , ] : H ×G→ H ⊆ G. Let F → E → B be a fibration sequence.

The topological analogs of the normal subgroups are the the relative Samelsonproducts

πn(ΩF ;Z/prZ)⊗πm(ΩE;Z/prZ)[ , ]r−−−−−→ πn+m(ΩF ;Z/prZ)→ πn+m(ΩE;Z/prZ)

associated to the loops on the fibration sequence

ΩF → ΩE → ΩB.

Such a compression of the Samelson product is an easy consequence of thenaturality of the Samelson product. But the Lie algebra identities need care. If pis a prime greater than 3, we claim that the map of differential graded Lie algebrasπ∗(ΩF ;Z/prZ) → π∗(ΩE;Z/prZ) is an extended differential ideal in the sense ofthe next definition. It is remarkable that this is a consequence of the existence ofa natural Samelson product and of the Hilton-Milnor theorem.

Definition 16.1. Let L′ → L be a morphism of graded Lie algebras. We callL′ an extended ideal of L if there are two bilinear pairings (called Lie brackets):

[ , ] : L′ × L→ L′

[ , ] : L× L′ → L′

such that1) the diagram of Lie brackets commutes

L′ × L′ [ , ]−−−−−→ L′

↓ ↓L′ × L [ , ]−−−−−→ L′

↓ ↓L× L [ , ]−−−−−→ L

2) for all x, y, and z in the union of L′ and L,

[x, y] = −(−1)deg(x)·deg(y)[y, x]

[x, [y, z]] = [[x, y], z] + (−1)deg(x)·deg(y)[y, [x, z]].

An extended differential ideal L′ → L is a morphism of differential graded Liealgebras which is an extended ideal and such that the differential d is a derivationin the sense that, for all x and y in the union of L′ and L,

d[x, y] = [dx, y] + (−1)deg(x)[x, dy].

We use the method of universal models to show that the theory of relativeSamelson products is a consequence of the Hilton-Milnor theorem and of the factthat we already have a functorial theory of Samelson products.

We substitute S1 for the nonexistent P 1(Z/prZ) and recall that π1(ΩX;Z/prZ) =π1(X)⊗ Z/prZ. We shall show that


Theorem 16.2. If p is a prime greater than 3, then π∗(ΩF ;Z/prZ)→ π∗(ΩE;Z/prZ), ∗ ≥1 is an extended differential ideal.

Remark 16.3. If p = 3, the above is true except for an error of order 3 in theJacobi identity and an error of order 9 in the triple vanishing identity for an odddimensional class.

A universal 2-variable model for Samelson products is the bouquet ΣPn,m =ΣPn(Z/prZ) ∨ ΣPm(Z/prZ).

Given homotopy classes f : Pn(Z/prZ)→ ΩE and g : Pm(Z/prZ)→ ΩE, therespective adjoints f : ΣPn(Z/prZ)→ E and g : ΣPm(Z/prZ)→ E define a mapf ∨ g; ΣPn,m → E. If

ιn : Pn(Z/prZ)→ ΩΣPn(Z/prZ) ⊆ ΩΣPn,m

ιm : Pm(Z/prZ)→ ΩΣPm(Z/prZ) ⊆ ΩΣPn,m

denote the two standard inclusions, then

Ω(f ∨ g)∗([ιn, ιm]) = [f, g].

We adopt the shorthand Pn = Pn(Z/prZ).The universal 2-variable model for relative Samelson products is the sequence

of inclusion map and projection map ΣPn → ΣPn,m → Pm. Given homotopyclasses f : Pn → ΩF and g : Pm → ΩE, the respective adjoints f : ΣPn → E andg : ΣPm → E define a commutative diagram of maps

ΣPnf−→ F

↓ ↓

ΣPn,mf∨g−−→ E

↓ ↓ΣPm

g−→ B.

The standard method of replacing a map F : X → Y by a fibration is a

functorial factorization F : Xι−→ EF

π−→ Y where

EF = (x, ω) ∈ X × Y I | fx = ω(0)

π(x, ω) = ω(1),

ι(x) = (x, ωx) where ωx is the constant path atx.

Note that the inclusion X → EF is a strong deformation retraction and hencethe maps

X × [0, 1]→ X × [0, 1] ∪ EF × 0, 1 → EF × [0, 1]

are all homotopy equivalences.Applying this to ΣPn,m → Pm yields a fibration sequence Fn,m → En,m → Pm

and a factorization

ΣPn → Fn,mΨn,m−−−→ F

↓ ↓ ↓ ιΣPn,m

'−→ En,mΦn,m−−−→ E

↓ ↓ π ↓ τΣPm

=−→ ΣPmτg−−→ B.


The fact that the inclusion ΣPn,m → En,m is an equivalence implies that f ∨ gcan be extended to a map of fibrations Φn,m : En,m → E. The fact that ΣPn,m ×[0, 1] ∪ En,m × 0, 1 → En,m × [0, 1] is an equivalence implies that this extensionis unique up to fibre homotopy.

In the above, Ψn,m : Fn,m → F is the restriction of Ψn,m and extends f.Note the factorization

[ιn, ιm] = (Ωι) [ιn, ιm]r : Pn+m → ΩFn,m → ΩE

The Hilton-Milnor theorem says that ΩFn,m → ΩEn,m has a retraction and hencethe choice of the factorization [ιn, ιm]r is unique. Define

Definition 16.4. The relative Samelson product [f, g]r ∈ πn+m(Ωf, Z/prZ)is

[f, g]r = (ΩΨn,m)∗([ιn, ιm]).

We have compatibility with the preceding Samelson product, that is,

(Ωι)∗[f, g]r = [(Ωι)∗f, g].

Not always, but when ΩF → ΩE admits a retraction, this property alone is enoughto determine [f, g]r uniquely.

Similarly, given homotopy classes g : Pm → ΩE and f : Pn → ΩF withadjoints as before, we use the universal 2-variable model ΣPn,m → Pm with thechange to the homotopy class [ιm, ιn] = [ιm, ιn]r in ΩFn,m to define

[g, f ]r = (ΩΨn,m)∗([ιm, ιn]r).

Remark 16.5. The relative Samelson products

[ , ]r : πn(ΩF ;Z/prZ)× πm(ΩE;Z/prZ)→ πn+m(ΩF ; z/prZ)

[ , ]r : πm(ΩE;Z/prZ)× πn(ΩF ;Z/prZ)→ πn+m(ΩF ; z/prZ)

have been constructed to be natural with respect to loop maps of ΩF → ΩE → ΩB,that is, natural with respect to maps of the fibration sequences F → E → B.

Lemma 16.6.[g, f ]r = −(−1)nm[f, g]r.

Proof. Since ΩFn,m is a retract of ΩEn,m, an equation [ιm, ιn] = −(−1)mn[ιn, ιm]which is true in ΩEn,m implies that the corresponding equation [ιm, ιn]r = −(−1)nm[ιn, ιm]ris true in ΩFn,m Hence,

[g, f ]r = (ΩΨn,m)∗([ιm, ιn]r) = −(−1)nm(ΩΨn,m)∗([ιn, ιm]r) = −(−1)nm[f, g]r.

Lemma 16.7.β([f, g] = [βf, g] + (−1)n[f, βg].

Proof. The equation β([ιn, ιm]) = [βιn, ιm] + (−1)n[ιn, ιm] is true in ΩEn,m.Therefore, the equation β([ιn, ιm]r) = [βιn, ιm]r + (−1)n[ιn, ιm]r is true in ΩFn,m.Hence,

β[f, g]r = β(ΩΨn,m)∗([ιn, ιm]r) =

(ΩΨn,m)∗(β[ιn, ιm]r) = (ΩΨn,m)∗([βιn, ιm]r + (−1)n[ιn, βιm]r) =

= [βf, g]r + (−1)n[f, βg]r.


Similarly, the 3-variable models

ΣPn,n → ΣPn,n,m → ΣPm,

ΣPn → ΣPn,m,m → ΣPm,m,

ΣPn → ΣPn,m,q → ΣPm,q,

are used in the proofs of the following identities:

Theorem 16.8. If

f ∈ πn(ΩF ;Z/prZ), g ∈ πm(ΩE;Z/prZ), h ∈ πq(ΩE;Z/prZ),

then[f + f1, g]r = [f, g]r + [f1, g]r,

[f, g + g1]r = [f, g]r + [f, g1]r,

[f, [g, h]]r = [[f, g]r, h]r + (−1)nm[g, [f, h]r]r.

17. Samelson products over the loops on an H-space

If H ⊆ G is a normal subgroup of a group with an abelian quotient group G/H,then the commutator map factors as

[ , ] : G×G→ H ⊆ G.The analog of this for Samelson products is the following:

Suppose that Fι−→ E

π−→ B is a fibration sequence. There is a map of differentialgraded Lie algebras (Ωι)∗ : π∗(ΩF ;Z/prZ) → π∗(ΩE;Z/prZ), ∗ ≥ 1. If B is anH-space with multiplication µ, then the Lie bracket into π∗(ΩE;Z/prZ) compressesinto a bilinear pairing

[ , ]µ : π∗(ΩE;Z/prZ)⊗ π∗(ΩE;Z/prZ)→ π∗(ΩF ;Z/prZ).

The compression depends on the multiplication µ. The mere existence of the factor-ization follows immediately from the fact that ΩB is homotopy commutative andhence all Samelson products vanish in it. The hard part is to show that the Liealgebra identities hold, that is,

Theorem 17.1. Suppose that B is an H-space. If p is a prime greater than 3,then, with the exception of anti-commutativity, the map

(Ωι)∗π∗(ΩF ;Z/prZ)→ π∗(ΩE;Z/prZ)

is a strong extended differential ideal in the sense of the definition below. Theanti-commutativity is replaced by the twisted anti-commutativity,

[f, g]mu = −(−1)deg(f)deg(g)[g, f ]µT .

If the multiplication in B is homotopy commutative, then anti-commutativityis valid without the twisting.

If p = 3, all of the above, minus the Jacobi identity and the triple vanishingidentity, are satisfied.

Remark 17.2. Suppose B is a connected H-space which is localized at an oddprime p and such that the rational Pontrjagin ring H∗(B;Q) is graded commu-tative. If B has only finitely many nonzero homotopy groups, then Zabrodsky[32] has shown that the multiplication in B can be altered so that it is homotopycommutative. This result was loosely and incorrectly stated in the author’s book[24].


Definition 17.3. Let L′ → L be a morphism of graded Lie algebras. We callL′ a strong extended ideal of L if there is a bilinear pairing (called a Lie bracket):

[ , ] : L× L→ L′

such that1) the diagram of Lie brackets commutes

L′ × L′ [ , ]−−−−−→ L′

↓ ↓L× L [ , ]−−−−−→ L′

↓ ↓L× L [ , ]−−−−−→ L

2) for all x, y, and z in the union of L′ and L,

[x, y] = −(−1)deg(x)·deg(y)[y, x]

[x, [y, z]] = [[x, y], z] + (−1)deg(x)·deg(y)[y, [x, z]].

A strong extended differential ideal L′ → L is a morphism of differential gradedLie algebras which is a strong extended ideal and such that the differential d is aderivation in the sense that, for all x and y in the union of L′ and L,

d[x, y] = [dx, y] + (−1)deg(x)[x, dy].

If f, g ∈ π∗(ΩE;Z/prZ), the subscript µ on [f, g]µ ∈ π∗(ΩF ;Z/prZ) indi-cates that the factorization depends on the multiplication in the H-space B. Thedefinition is via a universal model

ΣPn,mf∨g−−→ E

↓ ↓ πΣPn × ΣPm

Φ−→ B

where the extension Φ is defined using the H-space multiplication

ΣPn × ΣPmf×g−−−→ B ×B µ−→ B.

Since the universal model in this case does not have a base which is an H-space,more care is needed in the use of the models. Further details can be found in thebook [24].

Example 17.4. Let x be an even degree element and consider the formula Fvalid in any differential graded Lie algebra

F d(adk−1(x)(dx)) =1

2

k−1∑j=1

(j, k − j)[adj−1(x)(dx), adk−j−1(x)(dx)].

The formula F is proved by induction on k. The proof uses the formula for binomialcoefficients (i− 1, j) + (i, j − 1) = (i, j).

Now suppose that d = β is the Bockstein and that p is a prime greater than 3.The above formula F is valid for Samelson products in π∗(ΩE;Z/prZ) since thisis a differential graded Lie algebra.

If, in addition, ΩF → ΩE → ΩB is the loops on a fibration sequence andβx ∈ π∗(ΩF ;Z/prZ), then the relative Samelson product of the previous section,


denoted here with a subscript [x, y]r ∈ π∗(ΩF : Z/prZ) to distinguish it from theusual internal product in π∗(ΩE;Z/prZ), allows us to interpret the formula F asa valid formula in π∗(ΩF ;Z/prZ), that is, we write

ad(x)(βx) = [x, βx]r, adk−1(x)(βx) = [x, adk−2(x)(βx)]r.

The same proof of formula F works for these relative Samelson products.Suppose that B is an H-space with multiplication µ but we require only that

x, βx lie in π∗(ΩE;Z/prZ). We can use the Samelson product of this section toshow that the formula F is valid in π∗(ΩF ;Z/prZ), that is, we write

ad(x)(βx) = [x, βx]µ, adk−1(x)(βx) = [x, adk−2(x)(βx)]mu = [x, adk−2(x)(βx)]r.

The same proof is valid as before since we never have to use the formula fortwisted anti-commutativity. In addition, note that, once the Samelson productshave length≥ 3, the product of this section becomes identical to the relative productof the previous section, for example,

[x, [x, βx]µ]µ = [x, [x, βx]mu]r.

In other words, just one use of the product depending on the multiplication µ of Bputs the product into the loops on the fibre, and after that, the products are thesame as the relative products.

18. Mod p homotopy Bockstein spectral sequences

Let p be a prime. Consider the cofibration sequence

S1 p−→ S1 β−→ P 2(Z/pZ)ρ−→ S2 p−→

S2 β−→ P 3(Z/pZ)ρ−→ . . . .

If X is any space, this leads to the long exact sequence

π1(X)p←− π1(X)

β←− π2(X;Z/pZ)ρ←− π2(X)

p←− π2(X)β←− π3(X;Z/pZ)

ρ←− . . . .

If X is an arbitrary space, we shall alter, shorten, and terminate this sequenceby

0← π2(X)⊗ Z/pZ ρ←− π2(X)p←− π2(X)

β←− π3(X,Z/pZ)ρ←− . . . .

This being done, it is a short exact sequence of groups. All the groups except forπ3(X;Z/pZ) are abelian. If p is odd, even this is abelian. But, in any case, theimage of ρ : π3(X)→ π3(X;Z/pZ) is a central subgroup.

If X = G is a group-like space, we shall alter, extend, and terminate thissequence by

0← π1(G)⊗ Z/pZ ρ←− π1(G)p←− π1(G)

β←− π2(G;Z/pZ)ρ←− π2(G)

p←−

π2(G)β←− π3(G;Z/pZ)

ρ←− . . . .Once again, we have a short exact sequence of groups. All the groups exceptpossibly for π2(G;Z/pZ) are abelian and, if p is odd, this is abelian also. And theimage of ρ : π2(G)→ π2(G;Z/pZ) is always a central subgroup.

We set A∗ = π∗(X). In the first case of an arbitrary space X, this is understoodto be 0 if ∗ = 1. In the second case of a group-like space X, it is understood to be0 if ∗ ≤ 0.


We set E∗ = π∗(X;Z/pZ). In the first case, this is understood to be 0 if ∗ ≤ 1and equal to π2(X)⊗Z/pZ if ∗ = 2. In the second case, this is understood to be 0if ∗ ≤ 0 and equal to π1(X)⊗ Z/pZ if ∗ = 1.

In either of the above situations we get an exact couple C, that is, two gradedmodules A,E and an exact triangle of homomorphisms

Aι−→ A

∂ j

E

where:ι = p : A = π∗(X)→ A = π∗(X) is multiplication by p, hence, has degree 0.∂ = β : E = π∗(X;Z/pZ)→ A = π∗−1(X) is the Bockstein, hence, has degree

−1.j = ρ : A = π∗(X) → E = π∗(X;Z/pZ) is the mod p reduction map, hence,

has degree 0.The exact couple is displayed as follows:

↓ ι ↓ ιj−→ E

∂−→ Aj−→ E

∂−→ Aι−→

↓ ι ↓ ιj−→ E

∂−→ Aj−→ E

∂−→ Aι−→

↓ ι ↓ ιj−→ E

∂−→ Aj−→ E

∂−→ Aι−→

↓ ι ↓ ιIn the above, a path which goes two steps to the right, one step down, and repeatsthis is exact.

The term E has a differential d : E → E which is defined by d = j ∂. We set

E′ = H(E, d) = Z(E, d)/B(E, d) = ker(d)/im(d).

This leads to the derived couple C ′ of C which consists of the two gradedmodules:

A′ = ιA = im(ι), E′ = H(E, d)

together with the maps

ι′ : A′ → A′. ι′(ιa) = ι2a,

j′ : A′ → E′, j′(ιa) = ja+ im(d),

∂′ : E′ → A′, ∂′(e+ im(d)) = ∂e.

In our case, ι and ι′ are both multiplication by p. The map j = ρ is reductionmod p and j′ = ρ p−1 is the reduction mod p of the class divided by p. Themap ∂ = β is the mod p Bockstein and ∂′ is the mod p Bockstein on a cosetrepresentative.

The definition of an exact couple and of its derived couple are due to Massey[16, 17]. We leave as an exercise:

Lemma 18.1. 1) The maps ι′, j′, ∂′ are well defined, that is, they are inde-pendent of choices of representatives and they land in the appropriate groups.

2) The derived couple C ′ is exact.


The sequence of successive derived exact couples C, C ′, C ′′, C ′′′, . . . defines aspectral sequence via

E1 = E, E2 = E′, E3 = E′′, . . .

with differentials dr : Er → Er

d1 = d = j ∂, d2 = d′ = j′ ∂′, d3 = d′′ = j′′ ∂′′, d4 = d′′′ = j′′′ ∂′′′, . . .and we have that

Er+1 = H(Er, dr).

It is sometimes convenient to define the successive derived couples in one stepas in MacLane’s book [14]:

Let C be an exact couple. Define couples Cr as follows:

Ar = im(ιr−1 : A→ A) = ιr−1A,

Zr = ∂−1(im(ιr−1)) ⊆ E,Br = j(ker(ιr−1)) ⊆ E,

Er = Zr/Br = ∂−1(im(ιr−1)/j(ker(ιr−1))

and mapsιr = ι : Ar → Ar, ιr(ι

r−1a) = ιra,

jr = j ι−(r−1) : Ar → Er, jr(ιr−1a) = ja+ j(im(ιr−1)),

∂r = ∂ : Er → Ar, ∂r(e+ j(im(ιr−1)) = ∂e.

Note thatB1 ⊆ B2 ⊆ · · · ⊆ Br ⊆ Br+1 ⊆ . . .⊆ Zr+1 ⊆ Zr ⊆ · · · ⊆ Z2 ⊆ Z1.

We leave as an exercise

Lemma 18.2. The maps ιr, jr, ∂r are well defined.

Note that C1 = C. We need:

Lemma 18.3. The couples Cr are all exact and Cr+1 is the derived couple ofCr.

Proof. Assume that Cr is exact. It is sufficient to show that Cr+1 is thederived couple of Cr.

Suppose e = e+Br is a coset in Er, that is,

Br = j(ker(ιr−1)),

ande ∈ Zr = ∂−1(im(ιr−1)).

The r−th differential is described as follows:

dr(e) = ja+ j(ker(ιr−1)) = ja, where ∂e = ιr−1a, a ∈ A.

First we determine the group of boundaries im(dr) ⊆ Er:

Claim:im(dr) = Br+1 +Br ⊆ E.

ιra = ι∂e = 0 and ja ∈ j(ker(ιr)) = Br+1. Thus,

im(dr) ⊆ Br+1 +Br.


On the other hand, ιra = 0 implies that ιr−1a = ∂e for some e. Hence, e ∈ Zr,dr(e) = ja, and Br+1 +Br ⊆ im(dr).

Next we determine the group of cycles ker(dr) ⊆ Er :

Claim:

ker(dr) = Zr+1 +Br.

Observe that dre = 0 if and only if ja ∈ Br. That is,

ja = jb, ιr−1b = 0

a− b = ιc,

∂e = ιr−1a = ιr−1(b+ ιc) = ιrc.

Thus, ker(dr) ⊆ Zr+1 +Br.

On the other hand, ∂e = ιra implies that dr(e) = jιa + Br = 0 + Br. Hence,Zr+1 +Br ⊆ ker(dr).

Therefore,

H(Er, dr) = Er+1.

Consider the diagrams

π∗(X;Z/prZ)ρ−→ π∗(X;Z/pZ)

↓ β β

π∗−1(X)pr−1

−−−→ π∗−1(X)ρ−→ π∗−1(X;Z/pr−1Z)

π∗+1(X;Z/pr−1Z)β−→ π∗(X)

pr−1

−−−→ π∗(X)β ↓ ρ

π∗(X;Z/pZ)

It follows that

Zr∗ = β−1(im(π∗−1(X)pr−1

−−−→ π∗−1(X)))

= β−1(ker(π∗−1(X)ρ−→ π∗−1(X;Z/pr−1Z)))

= ker(π∗(X;Z/pZ)β−→ π∗−1(X;Z/pr−1Z))

= im(ρ : π∗(X;Z/prZ) −→ π∗(X;Z/pZ))

that is to say, classes in Er are represented by mod p classes which are thereductions of mod pr classes.

Br∗ = ρ(ker(π∗(X)pr−1

−−−→ π∗(X)))

= ρ(im(π∗+1(X;Z/pr−1Z)β−→ π∗(X)))

= im(β : π∗+1(X;Z/pr−1Z) −→ π∗(X;Z/pZ))

that is to say, classes represent zero in Er if they are Bocksteins associated tothe short exact coefficient sequence

0→ Z/pZ → Z/prZ → Z/pr−1Z → 0.


Note that dr is defined by the relation

E1 ⊇ Zr → Er

↓ ∂ |A1 |dr↑ ιr−1 ↓A1 j−→ Zr → Er.

The domain of this relation is Zr = β−1(im ιr−1) and the indetermancy isBr = imβ : π∗+1(X;Z/pr−1Z)→ π∗(X;Z/pZ.)

This description of dr yields the following identification of the differential

Lemma 18.4. If there is a factorization

f = F ρ : Pn(Z/pZ)→ Pn(Z/prZ)→ X,

then we have a factorization

drf = F β ρ ρ :

Pn−1(Z/pZ)ρ−→ Pn−1(Z/prZ)

ρ−→ Sn−1 β−→ Pn(Z/prZ)F−→ X,

that is,drf = ρρβF = ρβF.

Proof. Combine the definition of dr by the above relation with the diagram

Xf←− Pn(Z/pZ)

β−→ Sn−1

F ↓ ρ ↓ pr−1

Pn(Z/prZ)β←− Sn−1 ρ←− Pn−1(Z/prZ)

ρ←− Pn−1(Z/pZ)

Lemma 18.5 (Cartan-Eilenberg [3]). Suppose that there is a commutative dia-gram with the bottom row exact:

W ↓ γ ε

Xα−→ Y

β−→ Z

Then β induces an isomorphism im(γ)/im(α)∼=−→ im(ε).

Hence,

π∗(X;Z/prZ) ↓ ρ pr−1

π∗+1(X;Z/pr−1Z)β−→ π∗(X;Z/pZ)

η−→ π∗(X;Z/prZ)

where p is the fake multiple yields the identification of Er and the differential dr

Theorem 18.6.

Er∗ = Zr/Br = im pr−1 : π∗(X;Z/prZ)→ π∗(X;Z/prZ)

with the differential dr having the lift to dr : Zr → Zr given by, if

f = ρ(F ) ∈ π∗(X;Z/pZ), F ∈ π∗X;Z/prZ),

thendr(f) = ρβF ∈ π∗−1(X;Z/pZ).


Remark 18.7. The immediate identification of Er is as the image of the fakemultiple, that is,

impr−1 = η ρ : πn(X;Z/prZ)→ πn(X;Z/pZ)→ πn(X;Z/prZ).

But we know that, no matter what the prime, the image of the fake multiple is thesame as the image of the true multiple in dimensions ≥ 3. When X is a loop space,this is true in dimensions ≥ 2.

In addition, if p is an odd prime, then the fake and true multiples are the samemaps in dimensions ≥ 4, and if X is a loop space, they are the same maps indimensions ≥ 3.

When X = G is a group-like space, in dimension two

Er = impr−1 : π2(G;Z/prZ)→ π2(G;Z/prZ)

is the image of the fake multiple. In this case, we are unable to reduce this to thetrue multiple.

Remark 18.8. The above theorem is valid for all spaces X in dimension twowith the understanding that π2(X;Z/prZ) = π2(X)⊗Z/prZ. In this case, we havethe factorizations

π3(X;Z/pr−1Z)β−→ Tor(π2(X), Z/pr−1Z) ⊆ π2(X)

ρ−→ π2(X)⊗Z/prZ ρ−→ π2(X)⊗Z/pr−1Z

and we have the diagram with an exact row

π2(X)⊗ Z/prZ ↓ ρ pr−1

Tor(π2(X), Z/pr−1Z) → π2(X)⊗ Z/pZ η−→ π2(X)⊗ Z/prZ

Thus, in dimension two,

Zr = imρ : π2(X)⊗ Z/prZ ⊗ π2(X)→ π1(X)⊗ Z/pZ

Br = imβ : π3(X;Z/pr−1Z)→ Tor(π2(X), Z/pr−1Z)→ π2(X)⊗ Z/pZ)

Er = Zr/Br = impr−1 : π2(X)⊗ Z/prZ → π2(X)⊗ Z/prZ

IfX = G is a group-like space, we revert to the usual π2(G;Z/prZ) = [P 2(Z/prZ), G]∗and adopt the convention that π1(G;Z/prZ) = π1(G)⊗Z/prZ. Then the theoremis valid in all dimensions including dimension one. The obvious variation of thepreceding remarks applies.

We now prove a universal coefficient exact sequence for the r−term of theBockstein spectral sequence Er.

Theorem 18.9. There is a short exact sequence

0→ pr−1(π∗(X)⊗ Z/prZ)ρ−→ Er∗

β−→ pr−1Tor(π∗−1(X), Z/prZ)→ 0

and the r−th differential βr is the composition

Er∗β−→ pr−1Tor(π∗−1(X), Z/prZ) −→ pr−1(π∗−1(X)⊗ Z/prZ)

ρ−→ Er∗−1

where the middle map is induced by the inclusion Tor(π∗−1(X), Z/prZ) ⊆ π∗−1(X).


Proof. Consider the diagram of universal coefficient sequences

0 → π∗(X)⊗ Z/prZ → π∗(X;Z/prZ) → Tor(π∗−1(X), Z/prZ) → 0↓ 1⊗ ρ ↓ ρ ↓ pr−1

0 → π∗(X)⊗ Z/pZ → π∗(X;Z/pZ) → Tor(π∗−1(X), Z/pZ) → 0↓ 1⊗ η ↓ η ↓ include

0 → π∗(X)⊗ Z/prZ → π∗(X;Z/prZ) → Tor(π∗−1(X), Z/prZ) → 0

The universal coefficient sequence that we desire says that the images of the 3columns form a short exact sequence:

0→ im1 → im2 → im3 → 0.

The only nontrivial part is the exactness in the middle. This is an easy conse-quence of the facts that the upper left hand corner map 1 ⊗ ρ is an epimorphismand that the lower right hand corner map include is a monomorphism.

The description of βr is just the fact that it is induced on the image by theusual Bockstein.

Remark 18.10. The above universal coefficient sequence is split if p is an oddprime or if p = 2 and r ≥ 2 since it is a short exact sequence of vector spaces.

Remark 18.11. The differentials in the mod p homotopy Bockstein spectralsequence determine the p−primary torsion in the integral homotopy groups π∗(X)in the following way.

Assume that π∗(X) is finitely generated in each degree and has a decompo-sition into cyclic summands with a set of cyclic generators xi, yj , zki,j,k withorder(xi) =∞, order(yj) = prj , and order(zk) = qk with qk relatively prime to p.

Then E1π(X)∗ = π∗(X;Z/pZ) contains the following elements which generate

it and, if p is odd, are a basis:1) xi, yj ε π∗(X;Z/pZ) such that xi ⊗ 1 = xi, yj ⊗ 1 = yj via the reduction

map.2) and σ(yj) ε π∗+1(X;Z/pZ) such that

β(σ(yj)) = prj−1yj ∈ TorZ(π∗(X), Z/pZ) ⊆ π∗(X).

The differentials are as follows:1) βs(xi) = βs(yj) = 0 for all 1 ≤ s <∞2) βs(σ(yj)) = 0 for all 1 ≤ s < rj and βrj (σ(yj)) = yj .3) if p is odd or r ≥ 2, Erπ(X)∗ has a vector space basis xi, yj , σ(yj), with

rj ≥ r.4) E∞π (X)∗ = Erπ(X)∗ for r sufficiently large and has a basis xi.

19. Samelson products in Bockstein spectral sequences

Theorem 19.1. Let X = ΩY be a group-like space and p an odd prime. TheSamelson product together with the Bockstein differential makes Er = Er(ΩY ) adifferential graded Lie algebra (minus the Jacobi identity and the triple vanishingidentity if p = 3).

Proof. Recall that Zr = imρ : π∗(ΩY ;Z/prZ)→ π∗(ΩY ;Z/pZ). We know


Lemma 19.2. The reduction map commutes with Samelson products and theinflation maps commute modulo a power of p, that is,

πn(ΩY ;Z/prZ)⊗ πm(ΩY ;Z/prZ)[ , ]−−−−−→ πm+n(ΩY ;Z/prZ)

↓ ρ⊗ ρ ↓ ρπn(ΩY ;Z/pZ)⊗ πm(ΩY ;Z/pZ)

[ , ]−−−−−→ πm+n(ΩY ;Z/pZ)↓ η ⊗ η ↓ pr−1η

πn(ΩY ;Z/prZ)⊗ πm(ΩY ;Z/prZ)[ , ]−−−−−→ πm+n(ΩY ;Z/prZ)

It follows that the Samelson product defines a pairing on the image of ρ,

[ , ] : Zr ⊗ Zr → Zr, x⊗ y 7→ [x, y].

This pairing satisfies all the Lie identities with the known restrictions when p = 3.Furthermore, this pairing is compatible with the following pairing on the image

of η ρ,

[ , ] : im η ρ⊗ im η ρ→ im η ρ, ηρx⊗ ηρy 7→ ηρ[x, y] = pr−1[x, y].

This last pairing is well defined since ηρx = pr−1x = 0 implies that

ηρ[x.y] = pr−1[x, y] = [pr−1x, y] = [0, y] = 0.

We use the fact that


↓ pr−1 ↓ ηρ ∧ 1


commutes up to composition with Whitehead products. When dimension oneis involved, we also use the commutativity of the corresponding diagrams wherePn(Z/prZ) and/or Pm(Z/prZ) are replaced by spheres.

Hence, there is a well defined Samelson product on

Er = Zr/Br = im η ρ : π∗(ΩY ;Z/prZ)→ π∗(ΩY ;Z/prZ)

which is covered by the Samelson product on Zr.

Lemma 19.3. The derivation formula is valid, that is,

dr[x, y] = [drx, y] + (−1)deg x[x.dry].

Proof. Let ρ : π∗(ΩY ;Z/prZ) → π∗(ΩY ;Z/pZ) be the reduction map, andlet β : π∗(ΩY ;Z/prZ) → π∗−1(ΩY ;Z/prZ) be the Bockstein. Suppose ρx =x, ρy = y.

Then the differential is represented by

dr[x, y] = ρβ[x, y] = ρ[βx, y] + (−1)deg x[x, βy] =

[ρβx, ρy] + (−1)deg x[ρx, ρβy] = [drx, y] + (−1)deg x[x, dry].


Remark 19.4. Each Zr = Zr(ΩY ) is the sub-algebra of E1 = E1(ΩY ) =π∗(ΩY ;Z/pZ) which is the image of the reduction map ρ : π∗(ΩY ;Z/prZ) →π∗(ΩY ;Z/pZ). The differential drx represented by ρβx where ρx = x.

It follows that, if ΩF → ΩE → ΩB is the loops on a fibration sequence, then

Zr(ΩF )→ Zr(ΩE)

is an extended ideal with drx represented by ρβx. Hence,

Er(ΩF )→ Er(ΩE)

is also an extended ideal with drx represented by ρβx.If, in addition, B is an H-space, then

Zr(ΩF )→ Zr(ΩE)

and

Er(ΩF )→ Er(ΩE)

are strong extended ideals with drx represented by ρβx.

In mod 3r homotopy, the Jacobi identity and the triple vanishing identity arenot universally valid, but we do have

Theorem 19.5. In the mod 3 homotopy Bockstein spectral sequence Er(ΩY )of a loop space,

[x, [y, z]] = [[x, y], z] + (−1)deg x deg y[y, [x, z]], r ≥ 2

and

[x, [x, x]] = 0, if deg x is odd, r ≥ 3.

Proof. Let

J (x, y, z) = [x, [y, z]]− [[x, y], z]− (−1)deg x deg y[y, [x, z]]

be the error in the Jacobi identity. We know that 3J (x, y, z) = 0.In

Er = im η ρ = pr−1 : π∗(ΩY ;Z/prZ)→ π∗(ΩY ;Z/prZ)

we have

[x, y] = 3r−1[x, y]]

where 3r−1x = x, 3r−1y = y.Hence,

[x, [y, z]] = 3r−1[x, [y, z]]

and

J (x, y, z) = 3r−1J (x, y, z) = 0, r ≥ 2.

The Jacobi identity implies that, if x has odd degree,

3[x, [x, x]] = 0.

Hence, the Jacobi identity modulo an error term of order 3 implies that 3[x, [x, x]]has order 3.

Therefore,

[x, [x, x]] = 3r−1[x, [x, x]] = 0, r ≥ 3.


Example 19.6. Let p be a prime greater than 3. Given x ∈ π2n(ΩX;Z/prZ)with Bockstein βx ∈ π2n−1(ΩX;Z/prZ), consider the elements

τk(x) = adpk−1(x)(βx),

σk(x) =1

2

pk−1∑j=1

(j, pk − 1)

p[adj−1(x)(βx), adp

k−1(x)(βx)].

We know that βτk(x) = pσk(x).Hence,

τk(x) ∈ kernel β = ρβ : π2pkn−1(ΩX;Z/prZ)→ π2pkn−1(ΩX;Z/pZ)

and there exists

τk(x) ∈ π2pkn−1(ΩX;Z/pr+1Z)

such that ρτk(x) = τk(x).We could also say that the element represented by τk(x) in Er has drτk(x) = 0

and survives to be represented by τk(x) in Er+1. Since τk(x) actually reduces modpr to τk(x) and not just to the same mod p reduction mod boundaries, this is asharper statement than just saying the element survives to Er+1.

20. Nonexistence of rational Peterson spaces

In this section, we show that there do not exist rational Peterson spaces Pn(Q),that is, there do not exist spaces which have a single nonvanishing integral coho-mology group isomorphic to the rational numbers. This result is due to Kan andWhitehead [12]. Here is their proof.

First, recall some algebra:

Lemma 20.1. If 0 → A → B → C → 0 is a short exact sequence of abeliangroups and D is any abelian group, then there is a long exact sequence

0→ Hom(C,D)→ Hom(B,D)→ Hom(A,D)→

Ext(C,D)→ Ext(B,D)→ Ext(A,D)→ 0.

Hence, if Ext(A,D) 6= 0 for a subgroup A of B, then Ext(B,D) 6= 0. And, ifHom(C,D) 6= 0 for a quotient group C of B, then Hom(B,D) 6= 0.

In the same spirit, if Ext(A,D) is uncountable for a subgroup A of B, thenExt(B,D) is uncountable. And, if Hom(C,D) is uncountable for a quotient groupC of B, then Hom(B,D) is uncountable.

We also have a dual exactness which can useful in computations

Lemma 20.2. If 0 → A → B → C → 0 is a short exact sequence of abeliangroups and D is any abelian group, then there is a long exact sequence

0→ Hom(D,A)→ Hom(D,B)→ Hom(D,C)→

Ext(D,A)→ Ext(D,B)→ Ext(D,C)→ 0.

Among the important characteristics of Hom and Ext are

Lemma 20.3. If A is a free abelian group or if B is a divisible abelian group,then Ext(A,B) = 0.


Lemma 20.4. If A = lim→An is a direct limit, then we have the inverse limit

Hom(A,D) = lim←Hom(An, D)

and the short exact sequence

0→ lim←

1Hom(An, D)→ Ext(A,D)→ lim←Ext(An, D)→ 0.

Recall the universal coefficient theorem for cohomology

Theorem 20.5. For all k ≥ 1, there is a short exact sequence

0→ Ext(Hk−1(X;Z), Z)→ Hk(X;Z)→ Hom(Hk(X;Z), Z)→ 0.

Since Hom(A,Z) has no divisible subgroups, we get

Corollary 20.6. If the reduced integral cohomology groups of X are all divisi-ble, then the groups Hom(Hk(X;Z), Z) = 0 and Hk(X;Z) = Ext(Hk−1(X,Z), Z)for all k ≥ 1.

We need

Theorem 20.7. If A is an abelian group with Hom(A,Z) = 0, then A is torsionfree and divisible if and only if Ext(A,Z) is torsion free and divisible.

Proof. Let d be any nonzero integer and let dA = a ∈ A |da = 0 be thed−torsion subgroup. Since dA and A/dA are both torsion groups,

Hom(dA,Z) = Hom(A/dA,Z) = 0.

(The second vanishing statement also follows from the fact that A/dA is a quotientgroup of A.)

The exact sequence

0→d A→ Ad−→ A→ A/dA→ 0

factors into two short exact sequences

0→d A→ A→ dA→ 0,

0→ dA→ A→ A/dA→ 0.

Applying Ext( , Z) to these two exact sequences yields exact sequences

0← Ext(dA,Z)← Ext(A,Z)← Ext(dA,Z)← 0,

0← Ext(dA,Z)← Ext(A,Z)← Ext(A/dA,Z)← 0

which splice together to give the exact sequence

0← Ext(dA,Z)← Ext(A,Z)d←− Ext(A,Z)← Ext(A/dA,Z)← 0.

It follows that

Ext(dA,Z) = Ext(A,Z)/dExt(A,Z), Ext(A/dA,Z) =d Ext(A,Z).

Since every nonzero torsion group D contains a cyclic subgroup, it follows that,for such groups, Ext(D,Z) = 0 if and only if D = 0.

So then A has no d−torsion, that is, 0 =d A if and only if

0 = Ext(dA,Z) = Ext(A,Z)/dExt(A,Z)

if and only if Ext(A,Z) is d divisible.


Likewise, A is d divisible, that is, 0 = A/dA if and only if

0 = Ext(A/dA,Z) =d Ext(A,Z)

if and only if Ext(A,Z) has no d−torsion.The result follows.

Recall that an abelian group is torsion free and divisible if and only if it is arational vector space.

Lemma 20.8. If A is a nonzero rational vector space, then Ext(A,Z) is un-countable.

Proof. It is sufficient to show that Ext(Q,Z) is uncountable.The exact sequence 0→ Z → Q→ Q/Z → 0 yields the exact sequence

0→ Hom(Q,Q)→ Hom(Q,Q/Z)→ Ext(Q,Z)→ 0.

Since Hom(Q,Q) ∼= Q is countable, it is sufficient to show that Hom(Q,Q/Z) isuncountable.

Since Q =⋃

1n!Z,

Hom(Q,Q/Z) = lim←Hom(

1

n!Z,Q/Z) = lim

←Z/n!Z

and this is clearly uncountable.

Theorem 20.9. If X is a connected space for which all of the reduced integralcohomology groups are rational vector spaces, then Hk(X;Z) = 0 for all k ≥ 1. Inparticular, there are no rational Peterson spaces Pn(Q).

Proof. We know that

Hom(Hk(X;Z), Z) = 0

and

Hk(X;Z) = Ext(Hk−1(X;Z), Z)

for all k ≥ 1.The groups Ext(Hk−1(X;Z), Z) are all rational vector spaces and hence, so

are the groups Hk−1(X;Z). If Hk−1(X;Z) 6= 0 for some k ≥ 1, then the groupExt(Hk−1(X;Z), Z) = Hk(X,Z) is uncountable. Hence Hk−1(X;Z) = 0 for allk ≥ 1. Thus, Hk(X;Z) = 0 for all k ≥ 1.

21. Nonfinitely generated coefficients

For X a pointed topological space and A a finitely generated abelian group andn ≥ 2,

πn(X;A) = [Pn(A), X]∗.

We shall show

Theorem 21.1. a) Let n ≥ 2, let X ∈ X be a space in the homotopy categoryof pointed spaces, and let F ∈ F be an abelian group in the category of torsion freeabelian groups. There is a natural isomorphism of functors of two variables

πn(X)⊗ F → πn(X;F ).


b) Let n ≥ 4, let X ∈ X be a space in the homotopy category of pointed spaces,and let A ∈ A be an abelian group in the category of category of abelian groups withno 2-torsion. There is a natural short exact sequence of functors of two variables

0→ πn(X)⊗A→ πn(X;A)→ Tor(πn(X), A)→ 0.

Remark 21.2. It is sufficient to prove the above theorem for finitely generatedcoefficient groups. Then

πn(X;A) = limπn(X;B)

the direct limit being taken over all the finitely generated subgroups B of A. Sincetensor and Tor commute with direct limits and since limits of exact sequencesare exact, we get the natural isomorphism in part a) and the natural short exactsequence in b).

Again, since limits of exact sequences are exact, we get the long exact sequencesof a fibration in the case of nonfinitely generated coefficients.

Proof. Let

F =⊕α

Z, G =⊕β

Z

be finitely generated free abelian groups. Since [Pn(F ), Pn(G)]∗ = Hom(G,F ), n ≥2, it follows that

πn(X;F ) = [Pn(F ), X]∗ =⊕α

πn(X) = πn(X)⊗ F

is a natural isomorphism of functors of two variables. Merely note that an element

f = (fαβ) ∈ Hom(G,F )

is defined by a matrix of integers and

f∗ = (f∗βα) :⊕α

Sn →⊕β

Sn, f∗βα = fαβ

is defined by the dual matrix.Now let f : A → B be a homomorphism of finitely generated abelian groups

with no 2 torsion and let h = (h0, h1) :

0 → F1d−→ F0

ε−→ A → 0↓ h1 ↓ h0 ↓ f

0 → G1d−→ G0

ε−→ B → 0

be a chain map of finitely generated free resolutions covering the homomorphismf .

The choice of the chain map of free resolutions covering f is unique up to achain homotopy H : F0 → G1, that is, any other choice of a chain map h = (h0, h1)is of the form

h0 = h0 + d H, h1 = h1 +H d.Now consider the maps of spaces

Pn−1(G0)d∗−→ Pn−1(G1)

ι−→ Pn(B)ε∗−→ Pn(G0)

d∗−→ Pn(G1)↓ h∗0 ↓ h∗1 ↓ f∗ ↓ h∗0 ↓ h∗1

Pn−1(F0)d∗−→ Pn−1(F1)

ι−→ Pn(B)ε∗−→ Pn(F0)

d∗−→ Pn(F1)


and the maps

H∗ : Pn−1(G1)→ Pn−1(F0), H∗ : Pn(G1)→ Pn(F0).

The vertical maps of spaces are all unique by strong coefficient functoriality.The maps ι are the standard maps in the cofibration sequences. The remaininghorizontal maps and the maps H∗ are unique by strong coefficient functoriality.

In the above diagram, from left to right, the first, third, and fourth squarecommute by strong coefficient functoriality. The second square commutes by theconstruction of maps of cofibration sequences.

Since n ≥ 4, all the above maps are suspensions and hence all compositionsdistribute over sums. It follows that the above squares all commute when the chainmap h is replaced by the chain map h.

Thus, the dual maps are unique, and they form the commutative diagrambelow, even if we replace h by the chain homotopic map h:

πn−1(X;F0)d∗←− πn−1(X;F1)

ι∗←− πn(X;A)ε∗←− πn(X;G0)

d∗←− πn(X;G1)↓ h0 ↓ h1 ↓ f ↓ h0 ↓ h1

πn−1(X;G0)d∗←− πn−1(X;G1)

ι∗←− πn(X;B)ε∗←− πn(X;G0)

d∗←− πn(X;G1)

The horizontal arrows are exact and thus give the unique natural maps of shortexact sequences, that is, the maps of tensor and tor are the usual maps of tensorand derived functors, independent of the choice of the chain map h:

0 ← Tor(πn−1(X), A) ← πn(X;A) ← πn(X)⊗A → 0↓ f∗ ↓ f∗ ↓ 1⊗ f∗

0 ← Tor(πn−1(X), B) ← πn(X;B) ← πn(X)⊗B → 0

The uniqueness implies that compositions give the maps of exact sequencesassociated with compositions. Hence the universal coefficients sequence is a functor.

Example 21.3. Let R ⊆ Q be any subring of the rationals, for example, R = Q,R = Z(p), or R = Z[1/p] where p is a prime. Then, for n ≥ 2, the above theorempermits us to define

πn(X;R) = limπn(X;Z)

wherea) if R = Q, the limit is taken with respect to the maps n : Z → Z correspond-

ing to the subgroupsZ

(n− 1)!⊆ Z

n!.

b) if R = Z(p), the limit is taken with respect to the maps q : Z → Z corre-sponding to the subgroups

Z

r⊆ Z

qrwith r and q relatively prime to p.

c) if R = Z[1/p], the limit is taken with respect to the maps p : Z → Zcorresponding to the subgroups

Z

ps⊆ Z

ps+1.

And, in all these cases, the isomorphism πn(X;R) ∼= πn(X)⊗R is valid.For all of these, we have well known localized Hurewicz theorems [24]:


Theorem 21.4. : If X simply connected, n ≥ 2, and πi(X;R) = 0, i ≤ n−1,then πi(X;R) → Hi(X;R) is an isomorphism for i = n and an epimorphism fori = n+ 1.

The first question that arises in an attempt to prove a general Hurewicz theoremis:

For which abelian groups A and G is it true that

Tor(G,A) = G⊗A = 0

impliesH∗(K(G,n), A) = 0?

22. Computations with Hilton-Hopf invariants

Definition 22.1. If f : X → ΩY is a map to a loop space, then the multi-plicative extension is the composition of loop maps

f : ΩΣXΩΣf−−−→ ΩΣΩY

Ωe−−→ ΩY

where Σ : Y → ΩΣY, y 7→< , y > and e : ΣΩY → Y, < t, ω > 7→ ω(t) are thestructure maps of the adjoint functors Σ and Ω.

The commutative diagrams

ΣXΣ(Σ)−−−→ ΣΩΣX ΩΣΩX

Ωe−−→ ΣX 1 ↓ Σe ↑ Σ 1

ΣX ΩX

make it easy to check that

Lemma 22.2. f extends f in the sense that f Σ = f : X → ΩY and itis natural with respect to loop maps in the sense that (Ωg) f = Ωg f whereΩg : ΩY → ΩW.

Let f : ΣX → Z and g : ΣY → Z be maps with respective adjoints f : X → ΩZand g : Y → ΩZ. Of course, we have the defining compositions

f : XΣ−→ ΩΣX

Ωf−−→ ΩZ, f : ΣX

Σf−−→ ΣΩZe−→ Z.

Note that the multiplicative extension of f is the loop of the adjoint, that is,f = Ωf.

Definition 22.3. Recall that the commutator defines the Samelson product[f, g] : X∧Y → ΩZ and the adjoint is the Whitehead product [f, g] : Σ(X∧Y )→ Z.In particular, the multiplicative extension of the Samelson product is the loop ofthe Whitehead product.

We now describe the Hilton-Milnor theorem in one of its two adjoint forms.Let ωj(ι0, ι1) run over a basis of monomials for the free (ungraded) Lie algebra

L(ι0, ι1). For example, it could begin with

ω0(ι0, ι1) = ι0, ω1(ι0, ι1) = ι1, ω2(ι0, ι1) = [ι0, ι1]

ω3(ι0, ι1) = [ι0, [ι0, ι1]], ω4(ι0, ι1) = [ι1, [ι1, ι0]],

ω5(ι0, ι1) = [ι0, [ι0, [ι0, ι1]]], . . .


Let ι0 = ιX : XΣ−→ ΩΣX → ΩΣ(X∨Y ) and ι1 = ιY : Y

Σ−→ ΩΣY → ΩΣ(X∨Y )be the two inclusions.

Then each ωj(ιX , ιY ) : ωj(X,Y ) → ΩΣ(X ∨ Y ) is a Samelson product whereωj(X,Y ) is an appropriate smash product of X and Y . For example,

ω0(ιX , ιY ) = ιX : X → ΩΣ(X ∨ Y ), ω1(ιX , ιY ) = ιY : Y → ΩΣ(X ∨ Y ),

ω2(ιX , ιY ) = [ιX , ιY ] : X ∧ Y → ΩΣ(X ∨ Y ),

ω3(ιX , ιY ) = [ιX , [ιX , ιY ]] : X ∧ (X ∧ Y )→ ΩΣ(X ∨ Y ),

ω4(ιX , ιY ) = [ιY , [ιY , ιX ]] : Y ∧ (Y ∧X) :→ ΩΣ(X ∨ Y ),

ω5(ιX , ιY ) = [ιX , [ιX , [ιX , ιY ]]] : X ∧ (X ∧ (X ∧ Y )))→ ΩΣ(X ∨ Y ), , . . .

We state without proof the Hilton-Milnor theorem:

Theorem 22.4 (Hilton-Milnor). If X and Y are connected spaces, then thereis a natural homotopy equivalence

Ψ :∏j≥0

ΩΣ(ωj(X,Y ))→ ΩΣ(X ∨ Y )

where the left hand side is the weak product, that is, the direct limit of the finiteproducts, and we get the map Ψ by using the loop multiplication of ΩΣ(X ∨ Y )to multiply the multiplicative extensions of the above countable list of Samelsonproducts.

Remark 22.5. In order to get a map to the loop space defined on the weakproduct, the most convenient way to multiply maps is to require the j−th loop torun in the interval [1− 2j , 1− 2j+1].

Thus, we have a natural homotopy equivalence from the weak product to theloop suspension of the bouquet

Ψ : ΩΣX × ΩΣY × ΩΣ(X ∧ Y )× ΩΣ(X ∧X ∧ Y )× ΩΣ(Y ∧ Y ∧X)× . . .

→ ΩΣ(X ∨ Y ).

Let f : Z → ΩΣX be a map with adjoint f : ΣZ → X. Let µ : ΣX → ΣX∨ΣXbe the comultiplication of the suspension.

Definition 22.6 (Hilton-Hopf invariants). The k−th Hilton-Hopf invarianthk(f) of f is the projection on the k−th factor of the composition Ψ−1 Ωµ f ,that is,

Z → ΩΣXΩµ−−→ Ω(ΣX ∨ ΣX)

Ψ−1

−−−→∏j≥0

ΩΣ(ωj(X,X))πk−→ ΩΣωk(X,X).

Hence we have the Hilton-Milnor decomposition

Theorem 22.7 (Hilton-Milnor decomposition).

Ωµ(f) = ι0h0(f)× ι1h1(f)× [ι0, ι1]h2(f)× [ι0, [ι0, ι1]]h3(f)× . . .

The natural projections π0, π1 : ΣX ∨ ΣX → ΣX show that h0(f) = h1(f) = f .


Proof. One usesπ0 µ = π1 µ = 1ΣX

π0 ι0 = π1 π1 = 1ΣX , π0ι1 = π1 π0 = 0,

π0 ωj(ι0, ι1) = π1 ωj(ι0, ι1) = 0

for j ≥ 2 since, in these cases, ω(ι0, ι1) contains both ι0 and ι1.For example, applying Ωπ0 to the equation for Ωµ(f) yields

f = Ωπ0 Ωµ(f) = h0(f)× 0× 0× · · · = h0(f)

Since the only interesting Hilton-Hopf invariants hj(f) occur when j ≥ 2, weshall not refer to h0(f) = h1(f) = f as Hilton-Hopf invariants.

If Z is a finite dimensional complex, the above sums are finite. In fact, supposethat the dimension of Z is ≤ m− 1 and that X is n− 1 connected. Observe that,if m ≤ 2n, then hj(f) = 0 for all j ≥ 2, that is, all Hilton-Hopf invariants are 0.If m ≤ 3n, then hj(f) = 0 for all j ≥ 3, We shall be particularly interested in thelast case where the only nonzero Hopf invariant is

h2 : [ΣZ,X]∗ → [ΣZ,ΣX ∧X]∗

.

We now describe the Hilton-Milnor theorem in its equivalent adjoint form whichuses Whitehead products instead of Samelson products.

We shall abuse notation and use ι0 = ιX : ΣX → ΣX ∨ ΣY and ι1 = ιY :ΣY → ΣX ∨ΣY to denote the respective adjoints of the previous ι0 = ιX , ι1 = ιY .

Also, ωj(ιX , ιY ) : Σωj(X,Y ) → Σ(X ∨ Y ) is used to represent the Whiteheadproduct which is adjoint to the Samelson product.

Definition 22.8. The k−th Hilton-Hopf invariant

hk(f) : ΣZ → Σωk(X,X)

of the adjoint f is simply the adjoint of the previous Hilton-Hopf invariant.

Note that the comultiplication µ = ι0 + ι1 : ΣX → ΣX ∨ ΣX and the Hilton-Milnor decomposition can be written in the adjoint form where Whitehead productsreplace Samelson products. Observe that the adjoint of Ωµ(f) = Ω(ι0 + ι1) f is

(ι0 + ι1) f and the adjoint of ωj(ι0, ι1)hj(f) is ωj(ι0, ι1)hj(f). Hence,

Theorem 22.9 (Adjoint form of the Hilton-Milnor decomposition).

(ι0 + ι1) f = ι0 f + ι1 f + [ι0, ι1] h2(f) + [ι0, [ι0, ι1]] h3(f) + . . .

in the group [ΣZ,ΣX ∨ ΣX]∗. If Z is not itself a suspension, the order of theaddition may matter since this group may not be abelian .

The space ΣX∨ΣX is a univeral example for the Hilton-Milnor decomposition,that is, if g0, g1 : ΣX → W are any two maps, this defines a map g0 ∨ g1 :ΣX ∨ ΣX →W and we get

Corollary 22.10 (Distributivity formula).

(g0 + g1) f = g0 f + g1 f + [g0, g1] h2(f) + [g0, [g0, g1]] h3(f) + . . .

in the group [ΣZ,W ]∗. Again, if Z is not itself a suspension, the order of theaddition may matter.


The Hilton-Hopf invariants have two kinds of naturality

Lemma 22.11. 1) (right linearity) If h, k0, k1 : ΣT → ΣZ are maps, then

hj(f h) = hj(f) hhj(f (k0 + k1)) = hj(f) k0 + hj(f) k1

2) (left linearity with respect to suspensions) If

f : ΣZ → ΣX, Σ` : ΣX → ΣY

are maps, thenhj(Σ` f) = (Σ` ∧ · · · ∧ Σ`) hj(f).

The second part of the above lemma is a consequence of the naturality of theHilton-Milnor theorem for ΩΣ(A ∨B) with respect to maps of A and B.

We now apply the above to the study of compositions (g0 + g1) f wheref, g0, g1 : P 3(Z/prZ)→ P 3(Z/prZ). The distributivity formula is

(g0 + g1) f = g0 f + g1 f + [g0, g1] h2(f).

The Hilton-Hopf invariants hj(f) are zero for all j ≥ 3.

Let p : P 3(pr) → P 3(pr) be the fake multiple of the identity and let p :P 3(pr) → P 3(pr) be the actual multiple of the identity. Since both induce multi-plication by p on integral cohomology, their difference p − p = α induces zero oncohomology. Therefore, α factors as

α : P 3(pr)q−→ S3 α−→ S2 ι−→ P 3(pr).

We shall call such an α a fake zero.

Theorem 22.12. For all s ≥ 1,

ps = (p+ α)s = ps + α (ps−1(ps−1 + · · ·+ p+ 1)).

We need a sequence of lemmas:

Lemma 22.13 (composition of fake zeros is zero). If β and γ are fake zeros,then β γ = 0.

This is a consequence of the factorization of fake zeros,

β γ : P 3(Z/prZ)→ S2 ι−→ P 3(Z/prZ)q−→ S3 → P 3(Z/prZ).

Lemma 22.14. If β is a fake zero and f, g : P 3(Z/prZ)→ P 3(Z/prZ) are anymaps, then [f, β] h2(g) = 0. Hence, we have the distributive law

(f + β) g = f g + β g.

Proof. Since the dimension of P 3(Z/prZ) is 3, the Hilton-Hopf invariant fac-tors through the 3-skeleton as

h2(g) = (ι ∧ ι) G : P 3(Z/prZ)→ ΣS1 ∧ S1 → ΣP 2(Z/prZ) ∧ P 2(Z/prZ).

Hence

[f, β] h2(g) = [f, β] (ι ∧ ι) G = [f |S2 , β|S2 ] G = [f |S2 , 0] G = 0

since the fake zero β restricts to zero on S2.

The preceding two lemmas show that


Lemma 22.15. For fake zeros β and γ,

(f + β) γ = f γ.

Since the mod pr Hopf invariant is an isomorphism on fake zeros, Lemma 11.3,that is,

H(k β) = k2H(β) = H(β k2)

implies

Lemma 22.16. If β is a fake zero and k is an integer, then k β = β k2.

Corollary 22.17. For all s ≥ 1,

ps = (p+ α)s = (1 + α (ps−1 + ps−2 + · · ·+ p+ 1)) ps−1.

Proof. If p is an odd prime, the 3−rd homotopy groups are all abelian. Ifp = 2, we can at least say that 2 = 2+α is a central element in the group structure.

In any case, for any positive integer k,

(p+ α) k = (p+ α) + · · ·+ (p+ α) =p+ (p+ α) + α+ (p+ α) + · · ·+ (p+ α) =p 2 + α 2 + (p+ α) + · · ·+ (p+ α) =

p 2 + (p+ α) + α 2 + (p+ α) + · · ·+ (p+ α) =p 3 + α 3 + (p+ α) + · · ·+ (p+ α) = . . .

p k + α k

The inductive step to prove the corollary is

(p+ α) (ps + α (ps−1(ps−1 + · · ·+ p+ 1)) =(p+ α) ps + (p+ α) α (ps−1(ps−1 + · · ·+ p+ 1)) =

(ps + α ps) + p αps−1(ps−1 + · · ·+ p+ 1) =(ps+1 + α ps) + α p2ps−1(ps−1 + · · ·+ p+ 1) =

ps+1 + α ps(ps + · · ·+ p+ 1)

We claim that the elements α are divisible by p, more precisely:

Theorem 22.18. Let p be any prime and r ≥ 2. For all j ≥ 1, there exist fakezeros δj : P 3(Z/prZ)→ P 3(Z/prZ) which induce zero in integral cohomology suchthat α = δ1 p and

pj = pj + δj pj = (1 + δj) pj .

Proof. We know that both powers pr and pr are zero in π3(P 3(Z;Z/prZ);Z/prZ).Hence, 0 = αpr−1(pr−1 + · · ·+p+ 1). Since (pr−1 + · · ·+p+ 1) is relatively primeto p and this is an equation in the cyclic group K3(Z/prZ,Z/prZ) = Z/prZ, itfollows that α = δ p in this group.

For all j ≥ 1,

pj = (p+ α)j = pj + δ p (pj−1 + pj−2 + · · ·+ p+ 1)) pj−1 =

pj + δ (pj−1 + pj−2 + · · ·+ p+ 1) pj = pj + δj pj = (1 + δj) pj .


23. Cohomology of some cubic constructions

Let π be the cyclic group of order 3 with a generator σ. We let π act viaσ(x ∧ y ∧ z) = y ∧ z ∧ x on the smash product X ∧ X ∧ X. Let ω ∈ S1 be aprimitive cube root of unity. Then π acts on the right of S1 by multiplication,α ∗ σ = αω. Thus π acts on the left of the product W = S1 × (X ∧X ∧X, ∗) viaσ ∗ (α, x ∧ y ∧ z) = (αω−1, y ∧ z ∧ x).

Writing S1 = [1, ω] ∪ [ω, ω2] ∪ [ω2, 1] gives fundamental domains for the groupaction on W , that is, we can write

W = W0 ∪W1 ∪W2, Wi = [ωi, ωi+1]×X ∧X ∧X, i = 0, 1, 2.

Thus each Wi is a fundamental domain for the π space W and the orbit spacepair Γ1(X) = W/π = S1×π (X ∧X ∧X ∧X, ∗) can be written as the fundamentaldomain modulo an identification of the boundary.

W/π = W1/ ∼, (1, z) ∼ (ω, σz).

We define short filtrations of the orbit space pair by

G0 = 1, ω, ω2 ×π (X ∧X ∧X, ∗), G1 = Γ1(X) = S1 ×π (X ∧X ∧X, ∗).With mod 3 coefficients, these filtration lead to a homology spectral sequence

with

E10 = H∗(G0) = H∗(1, ω, ω2 ×π (X ∧X ∧X, ∗)) = 1⊗H∗(X ∧X ∧X, ∗),

E11 = H∗(G1, G0) = H∗(S

1 ×π (X ∧X ∧X∗), 1, ω, ω2 ×π (X ∧X ∧X, ∗)) =

H∗([1, ω]× (X ∧X ∧X, ∗), 1, ω × (X ∧X ∧X, ∗)) =

e1 ⊗H∗(X ∧X ∧X, ∗),with differential

d1(1⊗ x) = 0, d1(e1 ⊗ z) = 1⊗ (σz − z).Since the spectral sequence is confined to two lines, H∗(E

1, d1) = E2 = E∞.We can now compute the mod 3 homology and cohomology of the orbit space

pair Γ1(Pn(Z/3rZ)) = S1 ×π (Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), ∗). Recallthat we have the following mod 3 homology

H∗(Pn(Z/3rZ)) =< 1, un−1, vn >=< 1, u, v > .

Letx = v ⊗ u⊗ u, σx = u⊗ u⊗ v, σ2x = (−1)n−1u⊗ v ⊗ u

andy = u⊗ v ⊗ v, σy = v ⊗ v ⊗ u, σ2y = (−1)nv ⊗ u⊗ v.

Theorem 23.1.

H∗(S1 ×π (Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), ∗)) = H∗(Γ1(Pn(Z/3rZ))

has a basis

1⊗ u⊗ u⊗ u, 1⊗ v ⊗ v ⊗ v, 1⊗ x = 1⊗ σx = 1⊗ σ2x, 1⊗ y = 1⊗ σy = 1⊗ σ2y,

e1 ⊗ u⊗ u⊗ u, e1 ⊗ v ⊗ v ⊗ v, e1 ⊗ (1 + σ + σ2)x, e1 ⊗ (1 + σ + σ2)y.


Proof. The short filtration of the orbit space gives a spectral sequence. with

E10 = 1⊗H∗(Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), ∗),

E11 = e1 ⊗H∗(Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), ∗)

The first differential is d1(1⊗z) = 0, d1(e1⊗z) = 1⊗(σ−1)z. As a π module,there is a direct sum decomposition

H∗(Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), ∗) =< u⊗ u⊗ u > ⊕ < v ⊗ v ⊗ v > ⊕

< v ⊗ u⊗ u, u⊗ u⊗ v, u⊗ v ⊗ u > ⊕ < u⊗ v ⊗ v, v ⊗ v ⊗ u, v ⊗ u⊗ v > .

Since π acts trivially on the first two summands and freely on the last twosummands, it follows that E2 is given by the computation of the theorem. SinceE2 = E∞, the theorem follows.

We adopt the Milnor sign conventions for the actions of tensors. Thus

< α⊗ β ⊗ γ, x⊗ y ⊗ z >= (−1)βx+γx+γy < α, x >< β, y >< γ, z > .

Permutations act via adjoints, if

σ(x⊗ y ⊗ z) = (−1)z(x+y)z ⊗ x⊗ y,

then

< (α⊗ β ⊗ γ)σ, x⊗ y ⊗ z >=< α⊗ β ⊗ γ, σ(x⊗ y ⊗ z) >so that

(α⊗ β ⊗ γ)σ = σ−1(α⊗ β ⊗ γ) = (−1)α(β+γ)β ⊗ γ ⊗ α.

A dual spectral sequence to the above give the following cohomology compu-tation. Let H∗(Pn(Z/3rZ), ∗) =< µn−1, νn >, a = µ⊗ µ⊗ ν, b = ν ⊗ ν ⊗ µ.

Theorem 23.2.

H∗(S1 ×π (Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), ∗)) = H∗(Γ1(Pn(Z/3rZ))

has a basis

1⊗ µ⊗ µ⊗ µ, 1⊗ ν ⊗ ν ⊗ ν, 1⊗ a(1 + σ + σ2), 1⊗ b(1 + σ + σ2),

w ⊗ µ⊗ µ⊗ µ,w ⊗ ν ⊗ ν ⊗ ν, w ⊗ b = w ⊗ bσ = w ⊗ bσ2

where deg(w) = 1.

The proof is left as an exercise.

Let X be a pointed space and suppose µ ∈ Hn(X) is a class in mod 3 coho-mology. There is a functorial bundle pair

Γ(X) = Eπ ×π (X ∧X ∧X, ∗)→ Bπ

with fibre pair (X ∧ X ∧ X, ∗). This bundle pair leads to the definition of theSteenrod operations. First

Theorem 23.3. There is a canonical class

ξ(µ) = 1⊗ µ⊗ µ⊗ µ ∈ H3n(Γ(X))

such that the restriction to any fibre pair (X ∧X ∧X, ∗) is µ⊗ µ⊗ µ.


Proof. The mod 3 cohomology Serre spectral sequence has

Ep,q2 = Hp(Bπ;Hq(X ∧X ∧X, ∗).

This E2 term has local coefficients which reduce to the usual tensor product of baseand fibre cohomology when the π action on the fibre cohomology is trivial.

Let X = K = K(π, n) and let ι ∈ Hn(K(π, n)) be the universal example formod 3 cohomology, that is, there is a unique homotopy class f : X → K such thatf∗ι = µ. Since (K ∧ K ∧ K, ∗) is 3n − 1 connected with a π equivariant bottom

cohomology class ι⊗ι⊗ι, the class 1⊗ι⊗ι⊗ι ∈ E0,3n2 = H0(Bπ)⊗H3n(K∧K∧K, ∗)

survives to a well defined class ξ(ι) ∈ H3n(Γ(K)).Setting

ξ(µ) = f∗ξ(ι)

gives the result.

Recall the definition of the mod 3 Steenrod operations Pi on a class µ ∈ Hn(X).The reduced diagonal δ : X → X ∧ X ∧ X is π equivariant and thus defines theequivariant diagonal

δ = 1×π δ : Bπ × (X, ∗) = Eπ ×π (X, ∗)→ Eπ ×π (X ∧X ∧X, ∗).

The cohomology of the classifying space is

H∗(Bπ) = E(w)⊗ P (βw), deg(w) = 1, deg(βw) = 2.

We have

Definition 23.4. The mod 3 Steenrod operations Pi and their Bocksteins βPiare defined by

δ∗ξ(µ) = 1⊗D0 + w ⊗D1 + βw ⊗D2 + w(βw)⊗D3 + (βw)2 ⊗D4 + . . .

where Dj ∈ H3n−j(X, ∗) and there are fixed nonzero scalars εj such that

if j = 2(n− 2i), then Dj = εjPiµ

if j + 2(n− 2i)− 1, then Dj = εjβPiµ.Otherwise, Dj = 0 (For our purposes, the exact value of the fixed nonzero scalarsεj are unimportant.)

Thus the i−th mod 3 Steenrod operation Piµ on a class µ of degree n hasdegree n+ 4i.

.With mod 3 coefficients, H∗(Sn−1) =< 1, e > with degree e = n − 1 and

H ∗ (Pn(Z/3Z)) =< 1, µ, ν > with degree µ = n−1 and degree ν = n. We computethe mod 3 cohomology of the pairs

Γ(Sn−1) = Eπ ×π (Sn−1 ∧ Sn−1 ∧ Sn−1, ∗)

and

Γ(Pn(Z/3Z)) = Eπ ×π (Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z), ∗).

Theorem 23.5. a)

H∗(Γ(Sn−1)) = E(w)⊗ P (βw)⊗ e⊗ e⊗ e.


b)

H∗(Γ(Pn(Z/3Z))) =

E(w)⊗ P (βw)⊗ µ⊗ µ⊗ µ⊕Z/3Z ⊗ (1 + σ + σ2)(ν ⊗ µ⊗ µ)⊕Z/3Z ⊗ (1 + σ + σ2)(ν ⊗ ν ⊗ µ)⊕E(w)⊗ P (βw)⊗ µ⊗ µ⊗ µ

Proof. a) Since H∗(Sn−1 ∧ Sn−1 ∧ Sn−1, ∗) =< e ⊗ e ⊗ e > is a modulewith a trivial π action, the cohomology Serre spectral sequence of the fibration pairΓ(Sn−1)→ Bπ has

E2 = H∗(Bπ)⊗ e⊗ e⊗ e = E(w)⊗ P (βw)⊗ e⊗ e⊗ e.

Since it is confined to one horizontal line E∗,3n−32 , it collapses and E2 = E∞ with

no extension problems.b) The cohomology Serre spectral sequence of the fibration pair

Γ(Pn(Z/3Z))→ Bπ

is a spectral sequence of modules over the cohomology of the base H∗(Bπ) and hasE2 term given with twisted coefficients by

Ep,q2 = Hp(Bπ;Hq(Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), ∗)= Extp(Z/3Z,Hq(Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), ∗))

The resolution to compute

H∗(Bπ : M) = ExtZ/3Z[π](Z/3Z,M),M = H∗(X ∧X ∧X, ∗)is

Z/3Z[π]1−σ←−−− Z/3Z[π]

1+σ+σ2

←−−−−− Z/3Z[π]1−σ←−−− Z/3Z[π]

1−σ←−−− . . .We know the structure of H∗(Pn(Z/3rZ)∧Pn(Z/3rZ)∧Pn(Z/3rZ), ∗) as a π

module. It is a trivial π module in the dimensions 3n− 3 and 3n generated by theinvariant elements µ ⊗ µ ⊗ µ and ν ⊗ ν ⊗ ν, respectively. And it is a free modulein the dimensions 3n − 1 and 3n − 2 with generators α = ν ⊗ ν ⊗ µ, σα, σ2α andβ = ν ⊗ µ⊗ µ, σβ, σ2β, respectively.

Recall that group rings of finite groups over fields are Frobenius algebras andtherefore self-injective [7]. Hence, the above free modules are injective. It followsthat, when q = 3n− 1 or q = 3n− 2,Extp(Z/3Z,Hq(Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), ∗)) =Homπ(Z/3Z,Hq(Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), ∗)) =

invariants of Hq(Pn(Z/3rZ) ∧ Pn(Z/3rZ) ∧ Pn(Z/3rZ), p = 0

0, p 6= 0.

Hence, the nonzero degrees are

Ep,q2 =

E(w)⊗ P (βw)⊗ µ⊗ µ⊗ µ q = 3n− 3

Z/3Z ⊗ (1 + σ + σ2)(ν ⊗ µ⊗ µ) q = 3n− 2

Z/3Z ⊗ (1 + σ + σ2)(ν ⊗ ν ⊗ µ) q = 3n− 1

E(w)⊗ P (βw)⊗ µ⊗ µ⊗ µ q = 3n

(E2 is a module over H∗(Bπ) with w and βw annihilating the middle twoZ/3Z.)

The map Sn−1 → Pn(Z/3Z) induces a map from this spectral sequence in b) tothe previous spectral sequence in a). It follows that nothing on the line q = 3n− 3can be hit by a differential.


Since the spectral sequence is a module over H∗(Bπ) = E(w) ⊗ P (βw) withgenerators µ ⊗ µ ⊗ µ, (1 + σ + σ2)(ν ⊗ µ ⊗ µ), (1 + σ + σ2)(ν ⊗ ν ⊗ µ), ν ⊗ ν ⊗ νall on the line p = 0, it collapses at E2 = E∞. There can be no extension problemsand the result follows.

Eπ = S∞ is filtered by the π equivariant subspaces

S1 ⊂ S3 ⊂ S5 ⊂ . . . .

Hence, if M is a π module, the cohomology H∗(S1/π;M) is computed via theresolution

Z/3Z[π]1−σ←−−− Z/3Z[π].

We use the cohomology Serre spectral sequence to compute the cohomology of

Γ1(X) = S1 ×π (X ∧X ∧X, ∗)

in the two cases X = Sn−1 and X = Pn(Z/3rZ). We have E2 = H∗(S1/π;H∗(X ∧X ∧X, ∗) with local coefficients and get:

Theorem 23.6. a)

H∗(Γ1(Sn−1)) =< 1, w > ⊗e⊗ e⊗ e.

b)

H∗(Γ1(Pn(Z/3Z))) =

< 1, w > ⊗µ⊗ µ⊗ µ ⊕Z/3Z ⊗ (1 + σ + σ2)(ν ⊗ µ⊗ µ) ⊕w ⊗ ν ⊗ µ⊗ µ ⊕Z/3Z ⊗ (1 + σ + σ2)(ν ⊗ ν ⊗ µ) ⊕w ⊗ ν ⊗ µ⊗ µ ⊕< 1, w > ⊗µ⊗ µ⊗ µ

with (ν ⊗ µ⊗ µ) = σ(ν ⊗ µ⊗ µ) = σ2(ν ⊗ µ⊗ µ) in line 3 above and (ν ⊗ ν ⊗ µ) =σ(ν ⊗ µ⊗ µ) = σ2(ν ⊗ µ⊗ µ) in line 5 above.

Remark 23.7. In cohomology, the map Γ1(Pn(Z/3rZ)→ Γ(Pn(Z/3rZ) sendsmost of the generators to the identically named generators but it sends w⊗ν⊗µ⊗µ,w ⊗ ν ⊗ ν ⊗ µ and anything involving βw to zero.

We now compute the mod 3 Steenrod operation P1(1⊗ µ⊗ µ⊗ µ) in the mod3 cohomologies of Γ(Pn(Z/3Z)) and Γ1(Pn(Z/3Z)).

Theorem 23.8. a) In H∗(Γ(Pn(Z/3Z)),

P1(1⊗ µ⊗ µ⊗ µ) = aw ⊗ ν ⊗ ν ⊗ ν + b(βw)2 ⊗ µ⊗ µ⊗ µ.

where a 6= 0.b) In H∗(Γ1(Pn(Z/3Z)),

P1(1⊗ µ⊗ µ⊗ µ) = ±w ⊗ ν ⊗ ν ⊗ ν.

Proof. Note that part a) immediately implies part b).By the definition of the mod 3 Steenrod operations the equivariant diagonal

δ : Bπ × (Pn(Z/3Z), ∗)→ Eπ ×π (Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z), ∗)


induces the cohomology map

δ∗

: 1⊗ µ⊗ µ⊗ µ = ξ(µ) 7→ ε2n−2(βw)n−1 ⊗ µ+ ε2n−3w(βw)n−2 ⊗ ν.

The naturality of the Steenrod operation P1 and P1w = 0, P1(βw)k = k(βw)k+2

gives

δ∗

: P1(1⊗ µ⊗ µ⊗ µ) 7→ ε2n−2(n− 1)(βw)n+1 ⊗ µ+ ε2n−3(n− 2)w(βw)n ⊗ ν.

Hence, P1(1⊗ µ⊗ µ⊗ µ) 6= 0 and the only possibility is that

P1(1⊗ µ⊗ µ⊗ µ) = aw ⊗ ν ⊗ ν ⊗ ν + b(βw)2 ⊗ µ⊗ µ⊗ µ.

If a = 0, then the fact that δ∗

is a map of H∗(Bπ) modules shows that

δ∗

: P1(1⊗ µ⊗ µ⊗ µ) 7→ bε2n−2(βw)n+1 ⊗ µ+ bε2n−3w(βw)n ⊗ ν.

Hence, ε2n−2(n − 1) = bε2n−2, ε2n−3(n − 2) = bε2n−3 and b = n − 1 = n − 2which is impossible. Hence, a 6= 0.

24. Nonassociativity in smashes of mod 3 Moore spaces

In this section we show that the wedge decomposition of the smash product ofmod 3 Moore spaces is not associative. Hence, the Jacobi identity may not hold formod 3 homotopy. Unfortunately, in the original published version of this paper, Imade a mistake. I convinced myself that a modification of the same argument wouldshow the nonassociativity of the smash product of mod 3r Moore spaces for r ≥ 2.I falsely concluded that the Jacobi identity might also fail for homotopy modulo3r with r ≥ 2. This would have been a very subtle point since it was known thatthe Jacobi identity was valid in the mod 3 homotopy Bockstein spectral sequencefrom the E2 term onwards. But Brayton Gray pointed out the associativity ofwedge decomposition of mod 3r Moore spaces with r ≥ 2. We include in the nextsection his argument showing this and thus that the Jacobi identity is indeed validfor homotopy groups with coefficients mod 3r with r ≥ 2.

We begin by studying the failure of associativity for the decomposition of mod 3Moore spaces. In other words, we prove the nonassociativity of the comultiplicationmap for a smash product of mod 3 Moore spaces. This is the Spanier-Whiteheaddual of a fact discovered by Toda. [29] This argument appeared in this author’sthesis [20] but was never published.

Theorem 24.1. The maps

∆n,m : Pn+m(Z/3Z)→ Pn(Z/3Z) ∧ Pn(Z/3Z)

are not associative, that is, for some values of n,m, q ≥ 2, the diagrams

Pn+m+q(Z/3Z)∆n,m+q−−−−−→ Pn(Z/3Z) ∧ Pm+q(Z/3Z)

↓ ∆n+m,q ↓ 1 ∧∆m,q

Pn+m(Z/3Z) ∧ P q(Z/3Z)∆n,m∧1−−−−−→ Pn(Z/3Z) ∧ Pm(Z/3Z) ∧ P q(Z/3Z)

do not commute up to homotopy.


It suffices to treat the case when n = m = q.We reason by contradiction. Suppose that the above diagrams are all homotopy

commutative. Since the comultiplication map is homotopy commutative, it wouldfollow that, if ∆ = (1 ∧∆n.n) ∆n,2n, then

P 3n(Z/3Z)∆−→ Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z)

↓ 1 ↓ σP 3n(Z/3Z)

∆−→ Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z)

would homotopy commute for any even permutation σ, for example, for the cyclicpermutation σ(x ∧ y ∧ z) = z ∧ x ∧ y.

This follows from the homotopy commutative diagram

P 3n(Z/3Z)∆−→ Pn(Z/3Z) ∧ P 2n(Z/3Z)

1∧∆−−−→ Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z)↓ 1 ↓ 1

P 3n(Z/3Z)∆−→ P 2n(Z/3Z) ∧ Pn(Z/3Z)

∆∧1−−−→ Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z)↓ 1 ↓ T ↓ σ

P 3n(Z/3Z)∆−→ Pn(Z/3Z) ∧ P 2n(Z/3Z)

1∧∆−−−→ Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z)

(Note that the lower right hand square is actually strictly commutative.)

Let ∆ : P 3n(Z/3Z)→ Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z) be any map whichinduces an isomorphism in the top dimension 3n of mod 3 homology and thereforealso in mod 3 cohomology. Suppose that ∆ is homotopy equivariant in the sensethat ∆ is homotopic to the composition σ∆ with the cyclic permutation σ whereσ(x∧ y ∧ z) = z ∧ x∧ y. We shall show that this leads to a contradition. This gives

Corollary 24.2. The coproduct maps ∆ : Pn+m(Z/3Z) → Pn(Z/3Z) ∧Pm(Z/3Z) are not all associative. Hence, the Jacobi identity may fail for Samelsonproducts in mod 3 homotopy.

We proceed to derive the contradiction by constructing a π equivariant map

G : S1 × (P 3n(Z/3Z), ∗)→ S1 × (Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z), ∗)

where, if ω is a primitive cube root of unity, π =< 1, σ, σ2 > acts on S1 ×(P 3n(Z/3Z), ∗) via σ(α, x) = (αω−1, x) and acts on S1× (Pn(Z/3Z)∧Pn(Z/3Z)∧Pn(Z/3Z), ∗) via σ(α, x ∧ y ∧ z) = (αω−1, σ(x ∧ y ∧ z)).

Write S1 = [1, ω] ∪ [ω, ω2] ∪ [ω2, 1]. Suppose

F : [1, ω]× P 3n(Z/3Z)→ Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z)

is a homotopy with F (1, x) = ∆(x) and F (ω, x) = σ−1∆(x).Extend F to an equivariant map

F : S1 × P 3n(Z/3Z)→ Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z)

by requiring F (α, x) = σF (ασ−1, x).Now G is defined by G(α, x) = (α, F (α, x)).

Theorem 24.3. The induced map on orbit spaces

G : S1 ×π (P 3n(Z/3Z), ∗) = S1/π × (P 3n(Z/3Z), ∗)→

Γ1(Pn(Z/3Z)) = S1 ×π (Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z), ∗)


induces an isomorphism in the top dimension 3n+ 1 of mod 3 cohomology, that is,

∆∗(w ⊗ ν ⊗ ν ⊗ ν) = w ⊗ ν3n

.

Proof. Since the action of π on P 3n(Z/3Z) is trivial, the Kunneth theoremshows that H∗(S1 ×π (P 3n(Z/3Z), ∗) =< 1, w > ⊗ < µ3n−1, ν3n > .

The computation now follows from the fact that

∆ : S1 ×π (P 3n(Z/3Z), ∗)→ Γ1(Pn(Z/3Z))

is a bundle map which covers the identity on the base spaces S1 = S1/π → S1/πand is F (α, ) : P 3n(Z/3Z) → Pn(Z/3Z) ∧ Pn(Z/3Z) ∧ Pn(Z/3Z) on the fibreover [α]. Of course, all F (α, ) are homotopic to ∆ and induce a cohomologyisomorphism in dimension 3n.

25. Associativity in smashes of 3 primary Moore spaces

Let ∆ : Pn+m(Z/3rZ) → Pn(Z/3rZ) ∧ Pm(Z/3rZ) be the coproduct. Inthis section we prove it is associative if r ≥ 2 and n,m ≥ 3. That is, abbreviatePn(Z/3rZ) = Pn(3r) and consider the two maps

(∆ ∧ 1)∆ : Pn+m+q(3r)→ Pn+m(3r) ∧ P q(3r)→ Pn(3r) ∧ Pm(3r) ∧ P q(3r)

(1 ∧∆)∆ : Pn+m+q(3r)→ Pn(3r) ∧ Pm+q(3r)→ Pn(3r) ∧ Pm(3r) ∧ P q(3r).

Theorem 25.1. If m,n, q ≥ 3 and r ≥ 2 the above maps are homotopic.

Proof. Recall the coreduction maps ρ : Pn(3)→ Pn(3r) which are character-ized by the commutativity of

Sn−1 3−→ Sn−1 ι−→ Pn(3)q−→ Sn

↓= ↓ 3r−1 ↓ ρ ↓=Sn−1 3r−→ Sn−1 ι−→ Pn(3r)

q−→ Sn

Recall the diagrams

Pn+m(3)∆−→ Pn(3) ∧ Pm(3)

↓ ρ ↓ ρ ∧ ρPn+m(3r)

∆−→ Pn(3r) ∧ Pm(3r)

These diagrams are homotopy commutative modulo the addition of Whiteheadproducts. Since m,n ≥ 3, there can be no relavant Whitehead products and theyare in fact homotopy commutative. Hence, if D = (∆∧1)∆−(1∧∆)∆, the diagram

Pn+m+q(3)D−→ Pn(3) ∧ Pm(3) ∧ P q(3)

↓ ρ ↓ ρ ∧ ρ ∧ ρPn+m+q(3r)

D−→ Pn(3r) ∧ Pm(3r) ∧ P q(3r)is homotopy commutative.

Now suppose that

F : Pn+m+q(3s)→ Pn(3r) ∧ Pm(3r) ∧ P q(3r)

is any map which induces zero in mod 3r cohomology. Recall that Pn(3r)∧Pm(3r)∧P q(3r) ' Pn+m+q(3r) ∨ Pn+m+q−1(3r) ∨ Pn+m+q−1(3r) ∨ Pn+m+q−2(3r).


Lemma 25.2. a) F factors as

Pn+m+q(3s)F−→ Pn+m+q−2(3r)

ι−→ Pn(3r) ∧ Pm(3r) ∧ P q(3r)where ι is the inclusion of the wedge summand.

b) Any such map F factors as

Pn+m+q(3s)q−→ Sn+m+q γ−→ Sn+m+q−3 ι−→ Pn+m+q−2(3r)

where q is the pinch map and ι is the inclusion of the bottom cell.

Proof. a) The Hilton-Milnor theorem shows that the adjoint of F factorsthrough

ΩPn+m+q(3r)× ΩPn+m+q−1(3r)× ΩPn+m+q−1(3r)× ΩPn+m+q−2(3r).

The fact that F is zero in mod 3r cohomology implies that F factors throughPn+m+q−2(3r) since F cannot map nontrivially to any of the other summands.

b) It is elementary that F is null when composed with the pinch map q :Pn+m+q−2(3r)→ Sn+m+q2 . Hence, it factors through the fibre G of this map. Butin the relevant range of dimensions, G is homotopy equivalent to the bottom cellSn+m+q−3 ⊂ Pn+m+q−2(3r). Hence, F factors as

Pn+m+q(3s)→ Sn+m+q−3 ι−→ Pn+m+q2(3r).

Since all maps Sn+m+q−1 → Sn+m+q−3 are null at the prime 3, we get the requiredfactorization through the pinch map.

Use the above lemma to factor D : Pn+m+q(3)→ Pn(3) ∧ Pm(3) ∧ P q(3) andconsider the homotopy commutative diagram

Pn+m+q(3)q−→ Sn+m+q γ−→ Sn−1 ∧ Sm−1 ∧ Sq−1 ι−→ Pn(3) ∧ Pm(3) ∧ P q(3)

↓ 3r−1 ∧ 3r−1 ∧ 3r−1 ↓ ρ ∧ ρ ∧ ρSn−1 ∧ Sm−1 ∧ Sq−1 ι−→ Pn(3r) ∧ Pm(3r) ∧ P q(3r)

No matter what the parity of n+m+ q − 3, it follows that the compositions

(ρ ∧ ρ ∧ ρ) ·D = D · ρ : Pn+m+q(3)→ Pn(3r) ∧ Pm(3r) ∧ P q(3r)are null.

Since D factors through a wedge summand as

Pn+m+q(3r)D−→ Pn+m+q−2(3r) ⊂ Pn(3r) ∧ Pm(3r) ∧ P q(3r),

it follows that the composition D ·ρ : Pn+m+q(3)→ Pn+m+q(3r)→ Pn+m+q−2(3r)is null and D factors through the cofibre η, that is,

D = E · η : Pn+m+q(3r)→ Pn+m+q(3r−1)→ Pn+m+q−2(3r).

Since η is characterized by the commutative diagram

Sn−1 3r−→ Sn−1 ι−→ Pn(3r)q−→ Sn

↓ 3 ↓= ↓ η ↓ 3

Sn−1 3r−1

−−−→ Sn−1 ι−→ Pn(3r−1)q−→ Sn

and since E factors as

Pn+m+q(3r−1)q−→ Sn+m+q γ−→ Sn+m+q−3 ι−→ Pn+m+q−2(3r)

it follows that D = E · η = ι · γ · 3 · q and D are null.


It follows that

Theorem 25.3. If r ≥ 2 the Jacobi identity holds for Samelson products ofclasses of dimensions ≥ 3 in the mod 3r homotopy groups of a group-like space.

Since the reduction map ρ : π∗(G;Z/prZ) → π∗(G;Z/pZ) induces a surjec-tive morphism of Lie structures π∗(G;Z/prZ) → Er onto the mod p homotopyBockstein spectral sequence of a group-like space G, this gives an alternate proofof

Corollary 25.4. In the mod 3 homotopy Bockstein spectral sequence of agroup-like space, the Jacobi identity is valid from E2 onwards for classes of dimen-sions ≥ 3.

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Department of Mathematics, University of Rochester, Rochester, NY 14625

E-mail address: [email protected]

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