Embedding Certain Complexes up to Homotopy Type in Euclidean Space

Annals of Mathematics

Embedding Certain Complexes up to Homotopy Type in Euclidean SpaceAuthor(s): George CookeSource: Annals of Mathematics, Second Series, Vol. 90, No. 1 (Jul., 1969), pp. 144-156Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970685 .

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Embedding certain complexes up to homotopy type in euclidean space

By GEORGE COOKE*

Let Kbe a connected finite cw-complex. We say K embeds up to homotopy type in the n-sphere Sn if there is a subcomplex L of SI which is homotopy equivalent to K. Let a be an element of 1(Sm) Form the complex Snm - e". In this paper we study the question: For what k does Smn a-el embed up to homotopy type in sn+k+1?

Peterson obtained non-embedding results in [12] for complexes Stm e

where n - m = 2p - 2 for some prime p and a has non-zero mod p Hopf invariant. Peterson and Stein [13] obtained non-embedding results for a =

7)', a = V2. More general results were proved by Hilton and Spanier in [9]. First they showed that Snm - en always embeds in S+m+'. In order to state their non-embedding theorem we suppose that K is a subcomplex of Sn+k+l

and that K is homotopy equivalent to Stm - e". If k > 2, then the complement of K in Sn+k+l has the homotopy type of Sk -_ en+k-m for some,8 E 7rn+k-m-l(Sk).

(See the remarks at the beginning of the proof of Theorem 1 below.) If we are careful to orient everything properly, f8 is uniquely determined. The obvious question is: What is the relation between a and ,8? Hilton and Spanier proved that for sufficiently large p, SPa = ? Sp+m-kf, and so obtained their non-embedding result. (Compare Corollary 2 below.)

Finally, we mention a very general result of Stallings [14]. He proved that an (m - 1)-connected n-dimensional polyhedron embeds up to homotopy type in S2n-m+'. In particular, a complex of the form Snm-<ten embeds up to homotopy type in S2n-m+'. (This can also be deduced from the embedding result in [9], using the Freudenthal suspension theorem.)

The main result of this paper is as follows: Let Sm V Sk denote the one- point union (wedge) of Sm and Sk. Let i1: Sm C Sm V Sk and i2: Sk C Stm V Sk denote the inclusions.

THEOREM 1. Suppose n > m + 2, and k, m > 2. Then Smn-,aen embeds up to homotopy type in Sn+k+l with complement homotopy equivalent to Sk-i e +k-m if and only if there are elements q' in wr,1(Sm V Sk ) and * in Wn+kmn1(Sm V Sk) such that q' and * are in the cross-term (i.e., map to zero

* The author gratefully acknowledges that this work was supported in part by NSF grant GP-7952X.



EMBEDDING CERTAIN COMPLEXES 145

in the product Stn x Sk) and

[(io ? a) + ?, i2] = [(i2 0 a) + A, i1]

where [, ] is the Whitehead product as defined in [16].

In order to interpret the theorem we recall certain facts about the homotopy of SI' V Sk. The reader is referred to [8] for details. One chooses certain iterated Whitehead products of the generators i1: SI" C SI" V SI and

i2: Sk C SI' V Sk, called basic products. The pt" basic product is denoted by wG E wT~(Stn V SI). For example, A2 =i E Cm (Sm V Sk) I (& = 2 Ek(S V S ), and (w, = [il, i2] eWm+k-l(Sm V Sk). Then each basic product induces a homomorphism

((I)p)*: r*(SnI) r* (Sm V Sk)

For every p, (cop)* is a monomorphism, and wr*(Sn V Sk) is the direct sum of the images of (cop)*, p = -2, -1, 0, * .. Thus, given any element 7 E wi(Stm V Sk), y has a unique representation in the form

( 1) eY =Ep?-2 ((}Op 0 IY ) I 7E wri(SP)

The condition of the theorem asserts that there exist

9 = Ep2O (wi, a gP) , cpp e n-1(SPV) and

= Ep0 (OWP 0 *P) P Ip e En+k-mnl(S')

such that

(2) [(i ora) + p, i2] = [(i2 ofl) + A, i1] .

We expand both sides of (2), using standard formulas for rearranging White- head products and compositions. The basic rearrangement formula is given in Barcus and Barratt [2, p. 69]. (The reader should note that Barcus and Barratt use a definition of the Whitehead product that differs from White- head's definition by a change in sign.) Each side of (2) has a unique representation in the form (1), and so we obtain a sequence of relations equivalent to (2) by setting corresponding coefficients equal.

The first few relations so obtained are ( i ) Sk-la = (_r~kSxn-i2q

(ii) Sk lHa = (_)k+k+lSn-lo

(iii) SMHiS = (- 1)rn+lSk-1p0

Here S denotes the suspension operation, Si the i-fold iterate of S, and H the Hopf invariant.

COROLLARY 2. If S1-1,,el embeds up to homotopy type in Sn+k+1 with



146 GEORGE COOKE

complement homotopy equivalent to Sk Ad en+k-n , and m > 2, k > 2, n -m > 2, then Sk-la = ( 1)rnkSrnl 8fl

Corollary 2 implies that the theory of embeddings up to homotopy type differs markedly from the theory of PL-embeddings. If K is a polyhedron, then it is easy to prove that K PL-embeds in Sn if and only if its suspension SK PL- embeds in S+'. But the following example shows that this is not true for embeddings up to homotopy type.

Example. Let p be odd, and let c: SP SP denote the identity map. Suppose [c, c] E w2Pl(SP) is the i-fold suspension of a E w27Pli(SP-i) but is not an (i + 1)-fold suspension. (i is known for each p by Adams [1].) Further, suppose that p ? 1, 3, 7, 15, so that p > 2i + 1. Then we assert that SP-ia e2p-i does not embed in S2P+2. For, given an embedding of SP-i -ae2p- in S2p+2, Corollary 2 states that [c, c] = Sia = ? SP-i-'l,. But p - i - 1 > i,

which contradicts the hypothesis that [c, c] is not an (i + 1)-fold suspension.

On the other hand, S'(SP- -,,e2p-i) - Sp _-[,e2P embeds up to homotopy

type in S2p+2.

PROOF. Let i1: SP = SP x * c SP x SP and i2:SP = * x ScSP x SP denote

the inclusions. Then SP --[Ee2p is homotopy equivalent to the space obtained

from SP x SP by attaching a (p + 1)-cell via any map in the homotopy

class [i1] - [i2j. Now embed SP x SP x I in R2p+1. The homotopy class

i1] - [i2]: Sp - SP x SP x I contains an embedding j: SP ) SP x SP x I, by

a general position argument. Identify R2P+l with R2p+1 x 0 c R2p+2. Choose

any point v E R22 - R2 I'. The cone on j(SP) with vertex v is then embedded

in R2p+2 so that only its base j(SP) intersects R2p+'. The union of the cone

and SP x SP x I is then a subcomplex of R2p+2 homotopy equivalent to

It follows that there are arbitrarily large integers i such that there

exists a complex K and an integer n such that SiK embeds up to homotopy

type in Sn and yet no complex L such that SiK SiL and L embeds up to

homotopy type in Sn. When i = 1, or equivalently p = 4k + 1, we then obtain

counter-examples to a conjecture of Peterson (incorrectly stated in [12]): If

SX embeds up to homotopy type in Sn then there is a complex Y such that

(a) Y embeds up to homotopy type in S"',

(b) S Y Z SX.

Wu-Chung Hsiang has remarked to me that the manifolds which he

and Sczearba construct in [10] also provide counter-examples to Peterson's

conjecture.

COROLLARY 3. If a and ,3 are suspensions, and m > 2, k > 2, n - m > 2




then SI"--t en embeds up to homotopy type in Sn+k+1 with complement SkI-en+k- if and only if

Skl ow ( )nkSn-ljs

PROOF. Necessity follows from Corollary 2. To prove sufficiency, since a and ,8 are suspensions,

[(i, o a) + (, i2] = ( - 1)n+n-l[i1, i2] ? S 1I-a + [q', i2]

[(i2 / f) + *1 il] = ( - 1)n+m-1[i2, 1] ? Smn-1S + [*I il]

Apply Theorem 1 with = - = 0.

Proof of Theorem 1

We first make some remarks about orientation. We use cohomology throughout because of the application. The standard orientation on the unit ball B" c Rn is that generator u,, e Hn(B", S"n) given by the natural ordering of the n unit basis vectors in Rn. The standard orientation on the unit sphere S'-l is that generator vn-, E Hn-l(S'-l) such that av,-1 = (- 1)nun. Given a map f: S' - Stm, we orient the cells of the complex Sm -f en using the inclusion of the sphere Stn c Smn f en and the characteristic map (Bn, Sn-l) (S f e , Stm) of the nn-cell. If n - m ? 2, then these orientations give generators xm E Hm(Sm n- e") and xn e Hn(Sm--f en). It is an elementary homotopy theory result that if f, g: S?1-k Stn, then there is an orientation preserving homotopy equivalence Stn -f en > Stn m en if and only if f _ g.

Suppose that K is a subcomplex of Sn+k+l and that h: Sm n- en - K is a homotopy equivalence. Let U denote a closed regular neighborhood of K in Sn+k+l, and let i: K c U denote the inclusion. Then ih is a homotopy equivalence and we choose generators x' 1E Hm( U) and x' e Hn(U) such that

(ih)*x = xmn and (ih)*xn = Xnn Let V be the closure of the complement of U. Then U and V intersect

in their common boundary, a PL-manifold Mn+k. We choose a fundamental class [Ml E Hn+k(Mn+k) corresponding to the generator Vf"+k+l E Hn+k+l(Sn+k+l)

under the following sequence of isomorphisms

Hn+k(M) Hn+k+l(U. M) *-c. Hn k?1(Sn k?l, V) Hnk+(S n+k+)

Let ji: Ma U and j2: Ma V denote the inclusions. Then the Mayer- Vietoris sequence for the triad (Sn+k+l; U, V; M) shows that (j,)* and (j2)*

are monomorphisms and that

HP(M) j*HP(U)j j*HP(V), 0 < p < n + k.

Furthermore, by Poincare duality, the summands ji*Hv(U) and j*Hn+k-P(V)



148 GEORGE COOKE

are dually paired to the infinite cyclic group Hn+k(M) by the cup product. Thus there are unique classes y, e Hk(V) and Ynlkm e Hn+k -m(V) such that

(3) (j*Yn+k-m) -- (ijl*X) = (jl*xn) (j2Yk) = [M

We are assuming k > 2; a general position argument shows that V is simply- connected. The homology groups of V are infinite cyclic in dimensions 0, k, n + k - m, and zero otherwise. We want to say that V is homotopy equivalent to a complex of the form Sk g en+k-m. We use a result on cw- complexes. If K is a cw-complex, with skeletons {Ki, i = 0, 1, * * }, let C (K) denote the chain complex defined by CQ(K) = Hi(K', Ki'). The boundary operator in C(K) is the boundary operator of the triples (Ki, K"-', Ki-2). CQ(K) is free abelian, and if a is an i-cell of K then we denote by j the generator of CQ(K) corresponding to a.

THEOREM A. Let X be a connected, simply-connected space. Let C be a

chain complex of free abelian groups, generated by elements ca, indexed by an arbitrary set A. Assume that CO = Z, C1 = 0, and that there exists an

isomorphism q: H*(C) - H*(X) carrying the generator of Ho(C) = CO to the

generator of Ho(X) represented by the map of a one point space to X. Then there exists a cw-complex K with cells au, a E A, and a map f: Ka-y X such that

(a) The map *: C-o C(K) defined by +(ca) = rs is a chain isomorphism. (b) The diagram below is commutative.

\ / f*\ If*

H*(K)

Theorem A is a well-known, basic fact; it follows from the work of

J.H.C. Whitehead, Milnor and Wall, together with the proposition proved by Browder in [7, p. 28] which states that if C is a free chain complex, and C' is a chain complex, then any map H*(C) -- H*(C') is induced by some chain map C o C'. The reader is referred to [15, ?41 for a generalization of Theorem A to the non-simply-connected situation. For the convenience of the reader we include a sketch of a direct proof of Theorem A at the end of this paper.

We have chosen generators Yk, e Hk( Y) and Ynmk- e Hn+k-( Y). Since the homology groups are infinite cyclic, there are unique classes -k J H( Y)

and n?+k-rn C Hn+k-m( Y) such that <Yk, ok> = <yn+k-m, Yn+k-rn> = 1. With these

generators and the zero boundary operator H* ( Y) is a chain complex satisfying the hypotheses of Theorem A. Thus there is a complex Sk0 - en+k- m and a




homotopy equivalence h': SkI- en+k-m Y preserving orientation. The homotopy class of g is well-defined because of the remark at the end of the first paragraph of the proof of Theorem 1, and we denote it by (= E Wn+k-ml1(Sk).

Remark. The minus sign in (3) is necessary to ensure that if we now reverse the procedure, starting with the homotopy equivalence h', we will end up with the cohomology generators x' , xn in U and the homotopy class a for the attaching map.

The proof of Theorem 1 is based on a study of the homotopy type of the PL-manifold M =U v v. Since k and m are both at least 2, a general position argument shows that M is simply-connected. We choose generators for the cohomology of M by setting

z1 = 1 E H0(M) Z2 = 4 E Hm(M) Z3 = -j2 Yk E H (M) Z4 =jlxne H(M)

Z5 = 2 2Yn+k-m E Hn+k-(M) Z6 = [M] E H +k(M)

Then by (3) above we have

( 3') Z5 -z2= - Z4 - Z3 = Z6

By Theorem A above M is homotopy equivalent to a cw-complex with six cells, each carrying a cohomology generator corresponding to the appropriate generator in H*(M).

THEOREM 4. The PL-manifold M is homotopy equivalent to a cw- complex of the form

(Sm V Sk) ren en+k-m en+k

where the attaching maps y and j satisfy -i = (i1 o a) + q' and j = (i2 o f) + i

for some elements q' and r in the cross-term of wr*(Sn V Sk).

PROOF OF THEOREM 4. Assume without loss of generality that m < k. Then the cw-complex homotopy equivalent to M must in dimensions up through k be of the form

SIM eI if k < n,

S -e -- e if k >nX.

But since the inclusion M C U gives a map to St -f en of degree 1 in dimensions m and n, the attaching map for the k-cell is trivial.

Now let us write

M f (S V Sk) honmtp en+k-q en+k

In fact we can easily construct a homotopy equivalence



150 GEORGE COOKE

H: (Sm V S') "-- en -,5 en+k-m _- en+k M

such that j1(H I Sk): Sk U is null-homotopic. It follows that if we set

X = (Sm V Sk) -e', then the inclusion M C U induces a map of pairs

a: (X, Sm V Sk) (Snmm en, Sm) which on the subspace Sm V Sk simply pinches Sk to a point.

LEMMA 5. The attaching map 3 is homotopic to a map i into SF V Sk.

PROOF. Consider the diagram below

7rn+k-m(X, S-m

V VS)

> tt1m-l(S.

V Sk) > k-m Xl(X, STM V

Sk)

la* 11* 1(

zn a/,~~~~.. . (- e ) n!-k-m-I(SM) j 7rnn l-m-i(S--f en) > rn-- k-m-i(S- -is e,

Since the map X -Smm en extends over X,-^ en+km I3 maps to zero in S,-m en,

and hence to zero in (Smi--mf e", Sm). The map (X, Sm V Sk) (SIn e", Sm) in-

duces isomorphisms of relative homology groups, and so by the Blakers-Massey

theorem [4] it induces isomorphisms of relative homotopy groups in dimensions

less than n + k - 1. Since m > 2, the vertical maps at either end are both

isomorphisms. In particular, U3 maps to zero in (X, Sm V Sk), which proves

the Lemma. We complete the proof of Theorem 4 as follows. It follows by diagram

chasing that we can choose j E Wn+km1(S V Sk) mapping to 3 and satisfying

v()= 0. Thus 3 = (i20 o f') + +1 for some +1 in the cross-term. We show that

la' = A; the proof that a = (i, o a) + p is similar. Set Y = (Sm V Sk)- ,_en+k-m.

Then the inclusion MC V induces a map a': Y-Sk -5B en+k-m which is of

degree 1 on (nsk-m)-cells. Further, a' 5Sm V Sk is just the map to 5Sk

obtained by pinching S' to a point. We apply the elementary fact that if h

(X--f e, X) -

(Y - ge PI Y) is a map of pairs of degree 1, then h I Xmaps the

attaching map f to the attaching map g. Thus fi' = f8 and the proof of

Theorem 4 is complete.

Next we analyze the attaching map for the top cell in M. Set W =

(5m V k) -, en -.en"+k-m. Consider the diagram of inclusion maps below.

(y Sm v Sk) (XI Sm V Sk)

(W, Sm V Sk)

(W X) (WI Y)




The vertical maps induce isomorphisms of relative homology groups and so by the Blakers-Massey theorem they induce isomorphisms of relative homotopy groups in dimensions less than 2n + k - m - 2. Since n ? m + 2, 2n + k - m - 2 > n + k - 1, and so the "butterfly lemma" (see [17, p. 321) yields

(4) Wfn+k-1( W, St V Sk) -fl+k-1( Y, St V Sk) ( Wfl+k1 (X, Sm V Sk)

As we have remarked, the inclusions MC U and Mc V induce maps a: (X, St V Sk) (Sm __f e-, S-) and v': (Y. Sm V Sk) (Sk, g9 en+k-m, Sk). A refined Blakers-Massey theorem [5, p. 409] implies that in dimension n + k - 1 the homomorphisms of relative homotopy groups induced by a and a' are onto and describes the kernels in terms of the relative Whitehead product. The relative Whitehead product is a map wr(X, A) 0& wrj(A) L ']i+jwl(X, A) defined for any pair (X, A). See [3] for a definition. Let a': (B7, Sn-') _+ (X, Sm V Sk) and d': (Bn+k-m, Sn+k-m-l) - (Y, Sm V Sk) be characteristic maps for the cells attached by y and 3 respectively. Then it follows from the theorem mentioned above that

( i ) ker a. is cyclic, generated by [v', i2,] (ii) ker or' is cyclic, generated by [d', ilJ.

Now the inclusions M c U and M C V also induce maps

7: (W, Sm V Sk) o (Sm-f en, Sm)

and

7f: ( We S V Sk) > (Sk -- gen+k--m, Sk)

It follows from (4) that the direct sum -f A : Wl+k-1(W, Sm V Sk) ) w n+k4-S 'fe, Sf ) ED

n~~~~k-l(S~~ n km) e Wfn+k_(S -g en+k-m, Sk)

is onto, with kernel generated by [v', i 2J and [3', i1i. Here we use ar' and i' to denote characteristic maps in (W, Sn V Sk). Actually, [a', i21 and [i', ilJ generate infinite cyclic direct summands of Wfn+k1l(W, Sm V Sk), but we do not need that fact here.

Now let C e 7r.+k-l(W) be an attaching map for the (n + k)-cell of M. Let j: W - (W, Sm V Sk) denote inclusion. The inclusions of M _ U and M _ V induce extensions of z-: W -k S-m - fen and z': W -S 9 en+k-m over WI96en+k. Thus j*C e 7fn+k-l(W, Sm V -Sk) maps to zero in

wfn+k-l(S -f en, Sm) ( wfl+k _(Sk en+k-m, Sk)

We have proved that



152 GEORGE COOKE

( 5 ) j*C = r[jj', i2] + S[6', ill

for some integers r and s. The next lemma was made plausible to me by a result of James [11, p. 3771

which relates the cup product in a complex SIm - en - em+" to a certain relative Whitehead product.

Consider the cell complex W. Suppose 4 e ,,+k-1( W) is any element such that j.* = r[jj', i2J + s[6', ill e Wfn+k1l(W, Sm V Sk) for some integers r and s.

LEMMA 6. Choose a generator -

e Hn+k( W et+k) corresponding to the

fundamental class in (Bn+k, S?+k-1) under the characteristic map (Bn+k, S?+k-1) W e( W-c k W). Then the cup product in W en+k is given as follows:

Z15-Z2 = SZ6 9 Z4-Z3= rZ6

We defer the proof of Lemma 6 for a moment. We now complete the proof of Theorem 1, assuming Lemma 6. First, using (5) above, we may apply Lemma 6 with = C,9 6 = Z6. Then -r = s = 1 by our choice of generators made at the start of the proof (see (3) above). Thus

j*= [-I 9 i2] + [6', il].

Now in the exact sequence of the pair (W, Sm V Sk), aj*C = 0. But the relative Whitehead product satisfies

a[a, A]J = - [&ta, 19]

for any a e wT(X, A), 18 e wrj(A), [3, Eq. 3.5 p. 303]. Thus

aj* = [19 i2J - [6, ill = 0

This proves half of Theorem 1. On the other hand, assume there exist , ',

such that = (il o a) + p, = (i2 o 8) + * satisfy [1, i2] = [5, ilJ. Form the complex

W = (Sm v Sk) -r en _--en+k-,m

There are obvious maps z: W - >Sm _, en and z-': W Sk _ en+k-m. It follows

as before that

e* zD 75*: Wrn+k-(W Sn V Sk) 7 wn+k-l(S ae, SM)

e) wC+kJ(Skp ,e +k-, Sk)

is onto with kernel generated by [I', i2] and [3', ilJ, where y' and 3' are characteristic maps for the cells of W attached by -/ and 6. Since [, i2] =

[3, ilJ, the element -[', hi2 + [6' ill e n?+k-4 W1 S V Sk) vanishes under the boundary. We choose C e wT?+k-l(W) such that j*C = 9-[', i2] + [3', ilJ. The exact homotopy sequence for the pair (W, Sm V Sk) shows that we can vary




4 by the image of any element of Wfl+kl(Si V S'). It follows by diagram chase, since C already maps to zero in the relative groups Wfn+k-l(SM -L e~n, sM) and Wfl+k-1(Sk -, ek+%-, Sk), that we can vary C so that it maps to zero in Sm i -1en and Sk -en+k-i. Thus the maps z: W Sm---,a e and zI': W-e Sk-pen+k-m extend over W -en+k. By Lemma 6, W -en+k satisfies Poincare duality. The double mapping cylinder Z for the maps

W-a en+k Sm e

S k en+km

is simply-connected, and a homology sphere. Thus it is homotopy equivalent to Sk+n+'. We are therefore in a position to apply Browder's codimension one splitting theorem [6] to deduce that there is a PL-submanifold Mn+k c Sn+k+1

splitting the sphere into two pieces so that the triad (Sn+k+l; A, B; M) is homotopy equivalent to (Z; Sm - en , Sk _ en+k-m; W- e"+k). The hypotheses of Browder's theorem are all satisfied because dimension W - en+k = n + k > 6, all spaces involved are simply-connected, and the induced maps H2( W-aen+k

-+H2(Smn-. en)), H2(W-aen+k) H2(Sk-,p en+k-m) are clearly onto. This completes the proof of Theorem 1, assuming Lemma 6.

PROOF OF LEMMA 6. The proof is very similar to James's proof of his result in [11] referred to above. Consider the cell complex W = Y- e. Let en c W be an n-cell embedded in the interior of en. Pinching the boundary Sn- of eo to a point defines a map p: W-e W V Sn. Write

L = (Skg en+k-m) V Sn

and let F: W -+ L be the composition

W W rV

Sn (Sk en~k-m) V Sn = L

Then F is of degree 1 in cohomology in dimensions k and n (and n + k - m). Define z3 e Hk(L) and z' e H"(L) by F*z' = z3, F*z' = z4. The map F carries the subcomplex Sm V Sk of W to Sk 9 en+k-m in L and so induces a map of pairs which we denote by

F': (W. St V Sk) - (L, Sk _, gen+k-m)

Let w: (Bn, Sn-) + (L, Sk -gen+k-m) be the composition 7f ~~~F'

( B Sn-) ( We S V Sk) )(L9

Sk _g

en+k-,m

Then w is a characteristic map for the n-cell of L. By the naturality of the relative Whitehead product, F*[a', i2] = [W, i3J, where i3: Sk C L denotes in-

clusion. The map ij: Se- W, when composed with F, is null-homotopic and



154 GEORGE COOKE

so F*[6', ij] = 0. Thus

(6) Fj* = F*(r[-', i2] + s[6', ij1) = r[w, i3]

Let j': L (L, Sk'- en+k-m) denote inclusion. Then j*F*4 = r[w, i3J by (6). Let i4: St c L denote inclusion. Then

( 7) Wfn+kl(L) Wln+k-l(S ) @ 7Cn+k-1(S --ge + ) D Z

where the infinite cyclic summand is generated by the Whitehead product [i4, i3]. Now let (X, A) be any pair, and let i: A c X and j: X-+ (X, A) denote inclusions. Given a e wr,(X) and 13 e 7Tq(A), the relative Whitehead product satisfies

j*[a, i*/S] - [j~a, 13]

The reader is referred to [3, p. 306] for a proof. It follows that, in terms of the direct sum decomposition (7), (8) F*4 = x + y + r[i4, i3

for some x e 7rn+k-l(Sn) and y e wn+k_1(Sk- en+k-gm).

The map F induces a map

( 9) G: W-Z en+k > L- F - en+k

and we calculate the cup product in W-z en+k using the cohomology homomorphism induced by G. We have chosen generators z3 and z4 in H*(L). For any element 0 e 7n+k-l(L) the inclusion L c L '- en+k induces isomorphisms of cohomology groups in dimensions less than n + k, and we use z' and zo to denote also the corresponding elements in L '- en+k. We now use the following elementary fact about the relation between the Whitehead product and the cup product for the space L. If 0 e 7fl+k-l(L), then 0 = x + y + h[i4, i3J for some x e Wfn+k-1(Sn), ye W n+k1 (Sk -gen+k-,m), and integer h, according to the direct sum decomposition (7). Form the space L - en+k, and let u e Hn+k(L - e'+k ) be a generator corresponding to the fundamental class in (Bn+k, Sn+k-1) under the characteristic map for the cell en+k. The elementary result needed here is that in L-a n+k,

(10) - hu

(This follows very quickly from the fact that the Whitehead product in the wedge of the two spheres SI V Sq is the attaching map for the top cell of the product SI' x Sq.)

We apply (10) to the case where 0 = F*4. It follows from (8) that Zig-,Z = ru in L -F ozen+k. But the map G (see (9) above) is of degree 1 in dimension n + k, and so Z4'-Z3 = r-6 in WO-zen+k. This completes the proof of the first relation in Lemma 6. The other relation is proved in a similar way.




Sketch of proof of Theorem A

Suppose we have constructed the n-skeleton Kn with cells corresponding to the generators of C of degree at most n, and a map f.: K- X such that

( i ) Hi(fn) = 0 for i < n. (ii) The map A: C C(K1) defined in dimensions at most n is a chain

map. (iii) The diagram below is commutative for i < n.

'A Hi(C) Hi(X)

32*\, /(f n)*

Hi(KI)

(iv) Set f, = fn I K--'. Consider the square

l fn

f f1-1

Let (fn, 1)*: Hn(K', Kn-') Hn(fn-1) denote the induced homomorphism, and let i*: Hn(X) - H.(fn.-) denote the inclusion homomorphism. Write B., Zn for the boundaries and cycles in C., and let p: Zn, -+H,(C) denote the projection. Finally, assume that

(ll) (~~~fig 1)** 1Zn = i*g9Pp Zn Hn.(fn.,)

Assuming (i)-(iv), we proceed to the (n + 1)St stage as follows. B. is free abelian, and there is an exact sequence

Hn+l(fn) Hn (K , Ka - ) (fAs1*Hn(fn-,)

Equation (11) implies that (fi, 1)** is zero on B,. Thus there exists a homomorphism h: B, ) Hn+i(fn) such that aoh = *. Next, let Z,+1 be the cycles in C,+,, and define k: Z,+, - H.+1(f.) to be the composition

Zn+1 Zn+l/Bn+l =Hn+,(C) Hn+l(X)- Hn+,(fn) Now the sequence

0 - Zn+1 Cn+1 B- 0

is split exact; choose a splitting j: C,+,+ Z,+,. Finally, let g: Cn+1 Hn+1(fn) be the composition

jEa k+he Hn+l(hf) Cn+i > Z1+ (DB



156 GEORGE COOKE

Then g assigns to each c, generating Cn+1 an element g(cj) e H?+1(fn). Since K" is simply-connected (it has no 1-cells since C1 = 0), the Hurewicz homomorphism w~?1(f ) Hn+1(f fi) is an isomorphism by the relative Hurewicz theorem. Thus g(cj) e Hn+1(ff) is represented by an element g'(ca) e w~+(ff)

S?& > Kn'

g'(ca) { ]fn

Dn~l > X

Attach an (n + 1)-cell ua to Kn by ag'(c,,) and extend the map fi over Kn _A using g'(ca,). Proceed in this manner for each generator ca of Cn+1. Obtain a complex Kn+1 and a map fn+1: Kn+1 X such that (i), (ii), (iii), and (iv) are satisfied for n + 1.

INSTITUTE FOR ADVANCED STUDY, AND UNIVERSITY OF CALIFORNIA, BERKELEY.

BIBLIOGRAPHY

[ 1] ADAMS, J. F., Vector fields on spheres, Ann. of Math. 75 (1962), 603-632. [ 2] BARCUS, W. D., and BARRATT, M. G., On the homotopy classification of the extensions of

a fixed map, Trans. Amer. Math. Soc. 88 (1958), 57-74. [ 3] BLAKERS, A. L., and MASSEY, W. S., Products in homotopy theory, Ann. of Math. 58

(1953), 295-324. [4] , Homotopy groups of a triad: II, Ann. of Math. 55 (1952), 192-201. [5] , Homotopy groups of a triad: III, Ann. of Math. 58 (1953), 409-417. [6] BROWDER, W., Embedding 1-connected manifolds, Bull. Amer. Math. Soc. 72 (1966),

225-231. [7] , Torsion in H-spaces, Ann. of Math. 74 (1961), 24-51. [8] HILTON, P. J., Homotopy groups of the union of spheres, J. London Math. Soc. 30

(1955), 154-172. [9] and SPANIER, E. H., On the embeddability of certain complexes in euclidean

spaces, Proc. Amer. Math. Soc. 11 (1960), 523-526. [10] HSIANG, W-C., and SCZCARBA, R. H., On the embeddability and non-embeddability of

sphere bundles offer spheres, Ann. of Math. 80 (1964), 397-402. [11] JAMES, I. M., Note on cup-products, Proc. Amer. Math. Soc. 8 (1957), 374-383. [12] PETERSON, F. P., Some non-embedding problems, Bol. Soc. Mat. Mexicana (2) 2 (1957),

9-15. [13] and STEIN, N., The dual of a secondary cohomology operation, Ill. J. Math. 4

(1960), 397-404. [14] STALLINGS, J., The embedding of homotopy types into manifolds, mimeo notes, Princeton

University, 1965. [15] WALL, C. T. C., Finiteness conditions for Cw-complexes, Ann. of Math. 81 (1965), 56-69. [16] WHITEHEAD, J. H. C., On adding relations to hornotopy groups, Ann. of Math. 42 (1941),

409-428. [17] EILENBERG, S., and STEENROD, N. E., Foundations of Algebraic Topology, Princeton

University Press, Princeton, N. J., 1952.

(Received November 22, 1968)



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