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COSIMPLICIAL GROUPS AND SPACES OF HOMOMORPHISMS

bernardo villarreal herrera

Abstract: Let G be a real linear algebraic group and L a finitely generated cosimpli-cial group. We prove that the space of homomorphisms Hom(Ln, G) has a homotopystable decomposition for each n 1. When Ln = Fn/Kn where Fn is the free groupon n generators, Kn is a normal subgroup and G is a compact Lie group, we show thatthe decomposition of Hom(Fn/Kn, G) is G-equivariant.

The spaces Hom(Ln, G) assemble into a simplicial space Hom(L,G). When G = Uwe show that its bar construction B(L,U), has an E-ring space structure wheneverHom(L0, U(m)) is path connected for all m 1.

1 Introduction

The study of spaces of homomorphisms Hom(, G), where is a finitely generated groupand G a compact Lie group, has been quite rich in its own, and has set the foundations ofnew cohomology theories called q-nilpotent topological K-theory ([4]). The building blocksof these theories are the spaces Hom(Fn/

qn, U), where

qn is the q-th stage of the descending

central series of Fn, the free group on n letters, and U = colimm

U(m). If we fix q and let n

vary, these spaces form a simplicial space. Whenever a family {Ln}n0 of finitely generatedgroups fits into a cosimplicial group, it builds up a simplicial space Hom(L,G) induced by the

spaces of homomorphisms Hom(Ln, G). We show that the families {Fn/qn}n0, {Fn/F

(q)n }n0

where F(q)n is the q-th stage of the derived series of Fn, form a cosimplicial group, as well as

the surface groups {Tn}n1.When G is a real algebraic linear group, the simplicial space X = Hom(L,G) is sim-

plicially NDR, that is, if we set St(Xn) as the subspace of Xn in which any element is thecomposition of at least t degeneracy maps, then all pairs (St1(Xn), S

t(Xn)) are neighbor-hood deformation retracts. We prove this by showing that the subspaces St(Xn) are realsemi-algebraic sets for all 0 t n and therefore can be triangulated. As a consequence of[1], there are natural homotopy equivalences

Hom(Ln, G)

0kn

(Sk(Hom(Ln, G))/Sk+1(Hom(Ln, G))).

The free groups Fn, assemble into a cosimplicial group which we denote by F . In this caseHom(F,G) is NG, the nerve of G seen as a toplogical category with one object, which is alsothe underlying simplicial space of the classifying space BG. For each n, we take quotientsFn/Kn by normal subgroupsKn that are compatible with coface and codegeneracy maps of F

1

2 B. Villarreal

to get finitely generated cosimplicial groups which we denote by F/K. The induced simplicialspaces are more easily described since there is a simplicial inclusion Hom(F/K,G) NG.

Theorem 1.1. Let G be a real algebraic linear group and consider a compatible family ofnormal subgroups {Kn}n0 of F . Then there are natural homotopy equivalences

(n) : Hom(Fn/Kn, G)

1kn

(nk)

Hom(Fk/Kk, G)/S1(Fk/Kk, G)

,

where S1(Fn/Kn, G) := S1(Hom(Fn/Kn, G)).

The case when {Kn = qn} was first conjectured by Adem, Cohen and Gomez in [3], for

closed subgroups of GLn(C).

Corollary 1.2. If G is a Zariski closed subgroup of GLn(C), then there are homotopyequivalences for the cosimplicial group F/q,

(n) : Hom(Fn/qn, G)

1kn

(nk)

Hom(Fk/qn, G)/S1(Fn/

qn, G)

for all n and q.

Conjugation under elements of G gives Hom(Fn/Kn, G) a G-space structure and more-over, if G is a real algebraic linear group, then Hom(Fn/Kn, G) is a G-variety. The sub-spaces St(Hom(Fn/Kn, G)) are subvarieties that are invariant under the action of G forall 0 t n. Using techniques from [12], when G is a compact Lie group, we showthat Hom(Fn/Kn, G) has a G-CW-complex structure where each S

t(Hom(Fn/Kn, G)) is aG-subcomplex. This allows us to prove the equivariant version of the previous theorem.

Denote Rep(Fn/Kn, G) and S1(Fn/Kn, G) as the orbit spaces of Hom(Fn/Kn, G) andS1(Fn/Kn, G) respectively.

Theorem 1.3. Let G be a compact Lie group. Then (n) is a G-equivariant homotopyequivalence, and in particular we get homotopy equivalences

Rep(Fn/Kn, G)

1kn

(nk)

Rep(Fk/Kk, G)/S1(Fk/Kk, G)

.

In the second part of this paper we study the bar construction of Hom(L,G) for afinitely generated cosimplicial group, which we denote by B(L,G). For G = U we showthat B(L, U) has an I-rig structure, that is, if I stand for the category of finite sets andinjections, the functor B(L, U( )) : I Top is symmetric monoidal with respect to bothsymmetric monoidal structures on I. Using the machinery developed in [4] we prove:

Theorem 1.4. Let L be finitely generated cosimplicial group and suppose that the spaceHom(L0, U(m)) is path connected for all m 1. Then, B(L, U) is an E-ring space.

This theorem is also true if we replace U by SU, Sp, SO or O.

2 Homotopy Stable Decompositions 3

2 Homotopy Stable Decompositions

2.1 Spaces of Homomorphisms

Let G be a topological group and a finitely generated group. Any homomorphism : Gis uniquely determined by ((1), ..., (n)) G

n when 1, ..., n is a set of generators.On the other hand, if we fix a presentation of , then an n-tuple (g1, ..., gn) G

n will inducean element in Hom(, G) whenever {gi}

ni=1 satisfy the relations in the presentation of .

Thus, there is a one to one correspondence between the subset of such n-tuples in Gn andHom(, G). Topologize Hom(, G) with the subspace topology on Gn.

Lemma 2.1. Let : be a homomorphism of finitely generated groups. If G is atopological group, then : Hom(, G) Hom(, G) is continuous.

Proof. Suppose = a1, ..., ar | R and

= b1, ..., bm | R.

Recall that the induced map : Hom(, G) Hom(, G) is given by

((b1), ..., (bm)) 7 (((a1)), ..., ((ar)))

for : G. For any i, (ai) = bni1i1 b

niqiiqi

. By fixing one presentation for each (ai) weget that is given by

((b1), ..., (bm)) 7 ((bn1111

bn1q11q1

), ..., (bnr1r1 b

nrqrrqr )) =

((b11)n11 (b1q1 )

n1q1 , ..., (br1)nr1 (brqr )

nrqr ).

Therefore is the restriction of the map Gm Gr given by

(g1, ..., gm) 7 (gn1111 g

n1q11q1

, ..., gnr1r1 g

nrqrrqr )

which is continuous.

In particular, this Lemma tells us that given any two presentations of , we get anisomorphism : and hence a homeomorphism between the induced spaces ofhomomorphisms. Therefore the topology on the space of homomorphisms does not dependon the choice of presentations.

Recall that an affine variety is the zero locus in kn of a family of polynomials on nvariables over a field k. Throughout this paper we will focus only on k = R. If such anaffine variety has a group structure where the group operations are polynomial maps, i.e.,maps f = (f1, ..., fn) where each fi is a polynomial, it can be shown that this is in fact agroup of matrices. We call these groups linear algebraic groups. It is easy to check thatmatrix multiplication is in fact a polynomial map. For the inverse operation of matrices,we think real linear algebraic groups as subgroups of SL(n,R). Any matrix A in SL(n,R)satisfies A1 = Ct, the transpose of the cofactor matrix C of A. Since the cofactor matrix isdescribed only in terms of minors of A, the map A 7 Ct is a polynomial map.

4 B. Villarreal

Lemma 2.2. Let G be a linear algebraic group, then for any finitely generated group ,Hom(, G) is an affine variety. Moreover, if is a homomorphism of finitely generatedgroups, then is a polynomial map.

Proof. Suppose is generated by 1, 2, ..., r and has a presentation {p}. Each pis of the form n1k1

nqkq

= e, nj Z and kl {1, ..., r} for all 1 j q. For anyhomomorphism : G and any such relation p we have

(p) = (n1i1

nqiq) = (i1)

n1 (iq)nq = I,

the identity matrix in G. Since products and inverses in G are given in terms of polynomials,this sets up a family of polynomial relations {y,i,j},i,j, where each y,i,j is induced by(p)i,j = ij , the i, j entry of the matrix equality (p) = I. These relations do not dependon , only on p, in the sense that any r-tuple (g1, ..., gr) G satisfying {y,i,j},i,j, i.e.

(gn1i1 gnqiq)i,j = ij

for all and 1 i, j n, is an element of Hom(, G). Adding the polynomial relations{y,i,j} to the ones describing G

r define Hom(, G) as an affine variety.For the second part, recall from the proof of Lemma 2.1, that is defined in terms of

products and inverses of matrices and thus a polynomial map.

Similarly, this Lemma tells us that the affine variety structure on Hom(, G) does notdepend on the presentation of . Indeed, any isomorphism of groups will induce an isomor-phism of affine varieties.

2.2 Triangulation of Semi-algebraic Sets

Definition 2.3. A real semi-algebraic set is a finite union of subsets of the form

{x Rn | fi(x) > 0, gj(x) = 0},

where fi(x) and gj(x) are a finite number of polynomials with real coefficients.

Using Hilberts basis theorem, all real algebraic sets (affine varieties over R) are realsemi-algebraic sets. Indeed, the zero locus ideal of an affine variety will be finitely generatedand thus the affine variety can be carved out by finitely many polynomials.

What makes semi-algebraic sets more interesting is that images of semi-algebraic sets inRn under a polynomial map Rn Rm are semi-algebraic sets in Rm (see [6, p. 167]), as

opposed to affine varieties and regular maps.Let M , N be semi-algebraic subsets of Rm and Rn, respectively. A continuous map

f : M N is said to be semi-algebraic if its graph is a semi-algebraic set in Rm Rn. Thenext result is proven in [6].

Proposition 2.4. Given a finite system of bounded semi-algebraic sets Mi in Rn, there is

a simplicial complex K in Rn and a semi-algebraic homeomorphism k : |K| Mi where

each Mi is a finite union of k(int||)s with K.


Remark 2.5. Proposition 2.4 can be stated without the boundedness condition and thedetails can be found in [12], where they add the hipotesis

Mi closed in R

n .

In the next sections, we will be using this last result in its full extension, but a firstapplication is that any affine variety Z can be triangulated, that is, there exists a simplicialcomplex K and a homeomorphism |K| = Z. With Lemma 2.2 and Proposition 2.4 we provethe following.

Corollary 2.6. Let be a finitely generated group and G a real linear algebraic group.Then Hom(, G) is a triangulated space.

2.3 Simplicial Spaces and Homotopy Stable Decompositions

Let be category of finite sets [n] = {0, 1, ..., n} with morphisms f : [n] [m], orderpreserving maps. It can be shown that all morphisms in this category are generated bycomposition of maps denoted di : [n 1] [n] and si : [n + 1] [n] where 0 i n. Thismaps are determined by some relations between them, which are called simplicial identities.

For any category C, let Cop denote its opposite category. A functor

X : op Top

is called a simplicial space. Here Top stands for k-spaces, i.e., topological spaces where eachcompactly closed subset is closed. We denote Xn := X([n]) and the maps di = X(di) andsi = X(si) are called face and degeneracy maps respectively.

Fix n. Define S0(Xn) = Xn and for 0 < t n

St(Xn) =Jn,t

si1 sit(Xnt),

where sij : Xnj Xnj+1 is a degeneracy map, 1 i1 < < it n is a sequenceof t numbers between 1 and n, and Jn,t stands for all possible sequences. This defines adecreasing filtration of Xn,

Sn(Xn) Sn1(Xn) S

0(Xn) = Xn.

For each n there is a homotopy decomposition of Xn in terms of the quotient spacesSk(Xn)/S

k+1(Xn) with k n. To do this we need the following.Let A Z be spaces. Recall that (Z,A) is an NDR pair if there exist continuous functions

h : Z [0, 1] Z, u : Z [0, 1]

such that the following conditions are satisfied:1. A = u1(0),2. h(z, 0) = z for all z Z,3. h(a, t) = a for all a A and all t [0, 1], and4. h(z, 1) A for all z u1([0, 1)).

6 B. Villarreal

Examples of NDR pairs are pairs consisting of CW-complexes and subcomplexes. Indeed, ifZ is a CW-complex and A Z a subcomplex, then the inclusion A Z is a cofibrationwhich is equivalent to a retraction X I to A I X {0} relative to A {0}.

When X is a simplicial space, we call X simplicially NDR if

(St1(Xn), St(Xn))

is an NDR pair for every n and t 1. The following result can be found in [1].

Proposition 2.7. Let X be a simplicial space, and suppose X is simplicially NDR. Thenfor every n 0 there is a natural homotopy equivalence

(n) : Xn

0kn

(Sk(Xn)/Sk+1(Xn)).

For each n, the map (n) is natural with respect to morphisms of simplicial spaces, thatis, natural transformations X Y .

2.4 Homotopy Stable Decomposition of Hom(Ln, G)

Definition 2.8. Let Grp denote the category of groups. A functor L : Grp is calleda cosimplicial group. We say that L is a finitely generated cosimplicial group if each Ln isfinitely generated.

For any topological group G, its underlying group structure defines the functor

HomGrp( , G) : Grpop Set.

If L is a cosimplicial group, the composition of functors HomGrp( , G)L which we denoteby Hom(L,G) defines a simplicial set. Whenever L is finitely generated, for each n we cantopologize Hom(Ln, G) in a way that the induced face and degeneracy maps are continuous.Therefore we get the simplicial space Hom(L,G) : op Top.

Proposition 2.9. Let G be a real algebraic linear group, and L a finitely generated cosim-plicial group. Then for each n we have homotopy equivalences

Hom(Ln, G)

0kn

(Sk(Hom(Ln, G))/Sk+1(Hom(Ln, G))).

Proof. Fix n. Using Proposition 2.7, we only need to show that

(St1(Hom(Lk, G)), St(Hom(Lk, G)))

is a strong NDR-pair for all 1 t k n. To do this, first notice that each sj(Hom(Lk, G))is a semi-algebraic set, since Hom(Lk, G) is affine and sj is a polynomial map, for all 0 j, k n. Then, for all t 1 the finite union

St(Hom(Lk, G)) =i

si1 sit(Hom(Lkt, G))


is also a semi-algebraic set. Consider the natural filtration

Sk(Hom(Lk, G)) Sk1(Hom(Lk, G)) S

0(Hom(Lk, G)) = Hom(Lk, G).

The union

t St(Hom(Lk, G)) = Hom(Lk, G) is an affine variety, and therefore is a closed

subspace of some euclidean space. By 2.5, each St(Hom(Lk, G)) can be triangulated andtherefore is a CW complex. This way the inclusions St(Hom(Lk, G)) S

t1(Hom(Lk, G))are cofibrations and hence NDR-pairs. Therefore Hom(L,G) is simplicially NDR.

There are two canonical finitely generated cosimplicial groups which we will denote Fand F . These arise from finitely generated free groups.

Definition 2.10. Define F : Grp as follows: set F0 = {e} and for n 1 let Fn =a1, ..., an, the free group on n generators. The coface homomorphisms di : Fn1 Fn aregiven on the generators by

d0(aj) = aj+1

di(aj) =

aj j < iajaj+1 j = iaj+1 j > i

for 1 i n 1

dn(aj) = aj ;

and the codegeneracy homomorphisms si : Fn+1 Fn by

si(aj) =

aj j ie j = i+ 1

aj1 j > i+ 1

for 0 i n.

Here Hom(Fn, G) = Gn and the induced simplicial space Hom(F,G) is just NG the nerve

of G as a category with one object.

Definition 2.11. Define F : Grp as F n := a0, ..., an for any n 0; coface andcodegeneracy homomorphisms di : F n1 F n and si : F n+1 F n respectively, are given onthe generators by

di(aj) =

{aj j < iaj+1 j i

and si(aj) =

{aj j iaj1 j > i

for all 0 i n.

The induced simplicial space Hom(F ,G) = NG the nerve of the category G that has Gas space of objects and there is a unique morphism between any two objects.

Definition 2.12. We will say that a family of normal subgroups Kn Fn is compatible withF , if di(Kn1) Kn and si(Kn+1) Kn for all n and all i. Similarly we define compatiblefamilies of F .

8 B. Villarreal

Given {Kn}n0 a compatible family with F , we get induced homomorphisms

Fn1di //

Fn

Fn1/Kn1

di // Fn/Kn

Fn+1si //

Fn

Fn+1/Kn+1

si // Fn/Kn.

DefineF/K : Grp

as (F/K)n = Fn/Kn with coface and codegeneracy maps the quotient homomorphisms diand si respectively. This way F/K is a finitely generated cosimplicial group. Similarly, witha compatible family {Kn}n0 of F , we can define F/K : Grp.

Example 2.13. We describe two families of finitely generated cosimplicial groups that canbe constructed using F/K and F/K through the commutator subgroup. Let A be a group,define inductively 1(A) = A and q+1(A) = [q(A), A] for q > 1. The descending centralseries of A is

q(A) E E 2(A) E 1(A) = A.

Given a homomorphism of groups : A B, [a, a] = [(a), (a)] for all a, a in A, so that

(q(A)) q(B).

Taking A = Fn, and denoting qn :=

q(Fn), we have that the family of normal subgroups{qn}n0 is compatible with di and si. Thus we can define F/

q as (F/q)n = Fn/qn for all

q and 1 i n. Another example using the commutator is the derived series of a group A:

A(q) E E A(1) E A(0) = A

where A(i+1) = [A(i), A(i)]. Similarly, F/F (q), where (F/F (q))n = Fn/F(q)n defines a finitely

generated cosimplicial group.Similarly, by taking F

(q)n+1,

qn+1 F n, we also obtain the finitely generated cosimplicial

groups F/q+1 and F/F(q)+1.

Example 2.14. Now for n 0 consider the surface groups with presentations

Tn+1 = a0, ..., an, b0, ..., bn | [a0, b0] [an, bn] = 1.

Notice that the presentation of Tn is preserved by didi : F 2n1 F 2n+1 and sisi : F 2n+3 F 2n+1, where stands for the free product. Let p(Tk+1) be the normal subgroup of F 2n+1 suchthat T+1 = F 2+1/p(T+1). Then {p(Tk+1)}k0 is a family of compatible normal subgroupsof F F and thus, T+1 is a finitely generated cosimplicial group.

Remark 2.15. The symmetric groups n can not be assembled as a cosimplicial group.This is because there are no surjective homomorphisms n n1 for n 5. Indeed, givena homomorphism : n n1, ker is a normal subgroup of n, that is An or n. Thusthe image of is either the identity element or a subgroup of order 2.


For a cosimplicial group L, denote St(Lk, G) := St(Hom(Lk, G)).

Theorem 2.16. Let G be a real algebraic linear group and {Kn}n0 a compatible family ofnormal subgroups of F . Then there are natural homotopy equivalences

(n) : Hom(Fn/Kn, G)

1kn

(nk)

Hom(Fk/Kk, G)/S1(Fk/Kk, G)

.

Proof. Let 1 i1 < < ink n. Since Hom(Fn/Kn, G) Gn, we can consider the

projectionsPi1,...,im : Hom(Fn/Kn, G) Hom(Fnk/Knk, G)

given by (x1, ..., xn) 7 (xi1 , ..., xink). Assemble these maps together so that we get acontinuous map

n : Hom(Fn/Kn, G)Jn,k

Hom(Fnk/Knk, G)

given by(x1, ..., xn) 7 {Pi1,...,im(x1, ..., xn)}(i1,...,ink)Jn,k

where Jn,k runs over all possible sequences of length n k, 1 i1 < < ink n. Sinceall sequences (i1, ..., ink) Jn,k are disjoint, the restriction

n| : Sk(Fn/Kn, G)Jn,k

Hom(Fnk/Knk, G)

has a continuous inverse

Jn,ksj1 sjk where 1 j1 < < jk n and the intersec-

tion {j1, ..., jk} {i1, ..., ink} = . Therefore n| is a homeomorphism. Finally note thatSk+1(Fn/Kn, G) is mapped to

S1(Fnk/Knk, G). Taking quotients we get a homeomor-

phism

n| : Sk(Fn/Kn, G)/Sk+1(Fn/Kn, G)Jn,k

Hom(Fnk/Knk, G)/S1(Fnk/Knk, G)

The theorem then follows from Proposition 2.9.

The next corollary was first conjectured in [2, p. 12] for closed subgroups of GLn(C).Since any real linear algebraic group is Zariski closed we have the following version of theconjecture.

Corollary 2.17. If G is a Zariski closed subgroup of GLn(C), then there are homotopyequivalences for the cosimplicial group F/q,

(n) : Hom(Fn/qn, G)

1kn

(nk)

Hom(Fk/qn, G)/S1(Fn/

qn, G)

for all n and q.

10 B. Villarreal

Example 2.18. Let G = SU(2) and consider F/q. The case q = 2 (Fn/2n = Z

n) hasbeen largely studied, for example [3]. First a few preliminaries. Let T = S1 be a maximal

torus of G and W = N(T )/T = {[w], e} its Weyl group, where w =

(0 11 0

). W acts on

T via [w] t = wtw1 = t1 and using left translation on G/T we get a diagonal action onG/T T n. Let t = iR be the Lie algebra of T with the induced action of W . There is anequivariant homeomorphism t T {I} so that

G/T W (T {I})n = G/T W t

n.

The quotient map G/T (G/T )/W = RP2 is principal W -bundle and we can take theassociated vector bundle pn : G/T W t

n RP2. Let 2 be the canonical vector bundle overRP

2, then we can identify pn with n2, the Whitney sum of n copies of 2. The pieces inthe homotopy stable decomposition before suspending are

Hom(Zn, SU(2))/S1(Zn, SU(2)) =

{S3 if n = 1

(RP2)n2/sn(RP2) if n 2

where (RP)n2 is the associated Thom space of n2 and sn is its zero section.Let q = 3. An n-tuple (g1, ..., gn) lies in Hom(Fn/

3, SU(2)) if and only if [[gi, gj], gk] = Ifor all 1 i, j, k n, i.e., the commutators [gi, gj] are central in the subgroup generated byg1, ..., gn. Every non-abelian subgroup of SU(2) has center {I}, therefore [gi, gj] {I}for all 1 i, j n. Consider

Bn(SU(2), {I}) = {(g1, ..., gn) SU(2)n | [gi, gj] {I}}

the space of almost commuting tuples in SU(2). By the previous observation

Hom(Fn/3, SU(2)) = Bn(SU(2), {I}).

This spaces and the pieces of its homotopy stable decomposition were completely describedin [3]. We conclude

Hom(Fn/3, SU(2))/S1(Fn/

3, SU(2)) =

{S3 if n = 1

(

K(n)RP3+) (RP

2)n2/sn(RP2) if n 2

where K(2) = 1 and for n 3,

K(n) =7n

243n

8+

1

12.

Remark 2.19. For q 4 we can find nilpotent subgroups of SU(2) of class q. Indeed, if nis a representative in SU(2) of a primitive n-th root of unity, then the subgroup generated bythe set {2q, w} with w as above, is of nilpotent class q. With this we can show that the spacesHom(Fn/

q, SU(2)) for q 4 have more connected components than Hom(Fn/3, SU(2)).

More details in [5].


2.5 Equivariant Homotopy Stable Decomposition of Hom(Fn/Kn, G)

Definition 2.20. Let M be a G-space. We say that M has a G-CW-structure if thereexists a pair (X, ) such that X is a G-CW-complex and : X M is a G-equivarianthomeomorphism.

Consider the subspaces St(Fn/Kn, G) Hom(Fn/Kn, G). Under conjugation by elementsof G, these are G-spaces. We want to show that for all n, Hom(Fn/Kn, G) has a G-CWcomplex structure and all Sn(Fn/Kn, G) Sn1(Fn/Kn, G) Hom(Fn/Kn, G) areG-subcomplexes. To show this we slightly generalize some results in [12].

We continue using the techniques of the previous section, so we require G to be a reallinear algebraic group. It is known that any compact Lie group has a unique algebraic groupstructure (see [10, p. 247]). Assuming G is a compact Lie group, every representation spaceof G has finite orbit types (see [11]), so when M is an algebraic G-variety, the equivariantalgebraic embedding theorem [12, Proposition 3.2] implies that M has finite orbit types.Also, this theorem guarantees the existence of a G-invariant algebraic map f : M Rd forsome d such that the induced map f : M/G f(M) is a homeomorphism and f(M) is aclosed semi-algebraic set in Rd ([12, Lemma 3.4]). If : |K| M/G is a triangulation, wesay that is compatible with a family of subsets {Di} of M , if (Di) is a union of some(int||), where K and : M M/G is the quotient map.

Proposition 2.21. Let G be a compact Lie group, M0 an algebraic G-variety and {Mj}nj=1

a finite system of G-subvarieties of M0. Then there exists a semi-algebraic triangulation : |K| M/G compatible with the collection {Mj(H) | H is a subgroup of G}

nj=0 where

Mj (H) = {x Mj | Gx = gHg1 for some g G}.

Proof. Let H1, ..., Hs G be the orbit types of G on M and f : M Rd as above. By

[12, Lemma 3.3] allMj (Hi) are semi-algebraic sets, and therefore all f(Mj(Hi)) are also semi-algebraic. Since i, j vary on finite sets, we can use Proposition 2.5 and obtain a semi-algebraictriangulation

: |K| f(M) =ij

f(Mj(Hi))

such that each f(Mj(Hi)) is a finite union of some (int||), where K. Take =

f1 .

Proposition 2.22. Let G be a compact Lie group. Let M0 be an algebraic G-variety and{Mj}

kj=1 a finite system of G-subvarieties. Then M0 has a G-CW-complex structure such

that each Mj is a G-subcomplex of M .

Proof. Let : |K| M/G be as in Proposition 2.21 and : M M/G the orbit map.Let K be a barycentric subdivision of K, which guarantees that for any simplex n of K ,1((n n1)) Mj (Hn) for some Hn G and 0 j k. Since | :

1((n))/G

n is a homeomorphism and the orbit type of 1((nn1)) is constant, by [7, Lemma

12 B. Villarreal

4.4] there exists a continuous section s : (n) Mj so that s(nn1) has a constant

isotropy subgroup Hn. Consequently there is an equivariant homeomorphism

1(n n1) = G/Hn (n n1).

Collecting G-cells Gs (n) for all simplexes of K we get a G-CW structure over all Mj ,0 j k.

In a similar way as pairs of CW-complexes (X,A) are NDR pairs, we have that G-CW-complexes induce G-NDR pairs, i.e., NDR pairs with a G action where the representation(h, u) is equivariant ([0, 1] with the trivial action).

Lemma 2.23. Suppose that (X,A) is a G-NDR pair represented by (h, u). By passing toorbit spaces, the pair (h, u) induces an NDR representation (h, u) for the pair (X/G,A/G).Also, for any subgroup H G, by passing to fixed points (h, u) induces an NDR representa-tion (hH , uH) for the pair (XH , AH).

Proof. Since the representation (h, u) is equivariant, we have

X [0, 1] h //

X

X/G [0, 1] h // X/G

X

u // [0, 1]

X/G

u

;;

Conditions 1-4 of NDR pairs are easily checked since these are quotient maps induced byequivariant maps.

Similarly equivariance of (h, u) implies that the restrictions

X [0, 1] h // X

XH [0, 1]?

OO

hH // XH?

OO Xu // [0, 1]

XH?

OO

uH

3 B(L,U) as an E-ring space 13

Proof. We know that Hom(Fn/Kn, G) is an algebraic G-variety and for 1 j n thesubspaces Sj(Fn/Kn, G) are invariant under conjugation by elements of G. Assume G GLN (R). Each Sj(Fn/Kn, G) has at least j entries equal to the identity matrix I, thereforethese elements can be described as the zero set of polynomials of Hom(Fn/Kn, G) and theirintersection with the polynomials describing those j or more entries equal to I. Hence, by2.22 Hom(Fn/Kn, G) can be given a G-CW-complex structure where each Sj(Fn/Kn, G) is aG-subcomplex. Similarly, the quotient Hom(Fk/Kk, G)/S1(Fn/Kk, G) is a G-CW-complex.

To prove that the map (n) is a G-equivariant homotopy equivalence we need to showthat for any closed subgroup H G, the fixed point set map (n)H is a weak homotopyequivalence. To do this, consider

(n,H) : (Hom(Fn/Kn, G)H)

1kn

(nk)

Hom(Fk/Kk, G)H/S1(Fk/Kk, G)

H

induced by the natural transformation on the simplicial space Hom(F/K,G)H, which byLemma 2.23 and Proposition 2.7 is also a homotopy equivalence. The map (n)H agrees bynaturality with (n,H) and thus is a homotopy equivalence. The result now follows fromthe equivariant Whitehead Theorem.

Example 2.25. Let G = SU(2), and Kn = 2. Again, we follow the results in [3]. It was

proven that

Rep(Zn, SU(2))/S1(Zn, SU(2)) = Tn/W = Sn/2

where the action of the generating element on Sn is given by

(x0, x1, ..., xn) 7 (x0,x1, ...,xn).

Let Kn = 3. We know that Rep(Fn/

3, SU(2)) = Bn(SU(2), {I})/G and thereforethe stable pieces are given by

Rep(Fn/3, SU(2))/S1(Fn/

3, SU(2)) =

K(n)

S0

Sn/2.

where K(n) is the same as in the previous example.

3 B(L, U) as an E-ring space

3.1 The Bar Construction of Hom(L,G)

Definition 3.1. Let L be finitely generated cosimplicial group and G a topological group.Define

B(L,G) = |Hom(L,G)|.

14 B. Villarreal

For the cosimplicial groups F/q we get that B(F/q, G) = B(q, G), the classifying spacefor G-bundles of transitional nilpotency class less than q.

Let G,H be topological groups and f : G H a continuous homomorphism. If L is afinitely generated cosimplicial group then for each n, we have the commutative diagrams

Hom(Ln, G)di //

f

Hom(Ln1, G)

f

Hom(Ln, H)di // Hom(Ln1, H),

Hom(Ln, G)si //

f

Hom(Ln+1, G)

f

Hom(Ln, H)si // Hom(Ln+1, H)

for all 0 i, n, so that f is a simplicial map and therefore induces a continuous mapB(L, f) : B(L,G) B(L,H). By fixing a finitely generated cosimplicial group, L, we havea functor B(L, ) from the category of topological groups to Top. Conjugation by g Ginduces a continuous homomorphism G G. Hence B(L,G) is a G-space.

Lemma 3.2. Let L be a finitely generated cosimplicial group and G,H real algebraic lineargroups. Let p1 : GH G and p2 : GH H be the projections. Then

B(L, p1) B(L, p2) : B(L,GH) B(L,G) B(L,H)

is a natural homeomorphism.

Proof. Since GH is a direct product, p1 and p2 are continuous homomorphism and therefore

p = (p1) (p2) : Hom(L,GH) Hom(L,G)Hom(L,H)

is a simplicial map. Its easy to check that in fact is a simplicial isomorphism. Both G and Hbeing real algebraic, imply that Hom(Ln, G) and Hom(Ln, H) have a CW-complex structure,and therefore are k-spaces. By [8, Theorem 11.5] the composition

B(L,GH)|p| // Hom(L,G) Hom(L,H)|

|pi1||pi2|// B(L,G) B(L,H)

is a natural homeomorphism where |1 p| |2 p| = B(L, p1)B(L, p2).

3.2 Relation between commutative I-monoids and infinite loop spaces

In this section we recall briefly the notion of I-monoid and how it is related to infinite loopspaces. This is more widely covered in [4].

Let I stand for the category whose objects are the sets [0] = and [n] = {1, ..., n} foreach n 1, and morphisms are injective functions. Any morphism j : [n] [m] in I canbe factored as a canonical inclusion [n] [m] and a permutation m. This category issymmetric monoidal under two different operations, namely, concatenation [n][m] = [m+n]with symmetry morphism the permutation m,n n+m defined as

m,n(i) =

{n + i if i mim if i > m


and identity object [0]. The second operation is Cartesian product [m] [n] = [mn] withm,n mn given by

m,n((i 1)n + j) = (j 1)m+ i

where 1 i m and 1 j n. In this case the identity object is [1]. Cartesian product isdistributive under concatenation (both left and right).

Definition 3.3. An I-space is a functor X : I Top. This functor is determined by thefollowing.

1. A family of spaces {X [n]}n0, where each X [n] is a n-space;

2. n-equivariant structural maps jn : X [n] X [n+1] (here we consider X [n+1] is a n-space under the restriction of the n+1-action to the canonical inclusion n n+1)with the property: for any j : [n] [m], , m whose restrictions in n are equal,we have x = x X(j)(X [n]).

We say that an I-space X is a commutative I-monoid if it is a symmetric monoidal functorX : (I,, [0]) (Top,, {pt}). Additionally, we say that X is a commutative I-rig if X isalso symmetric monoidal with respect to (I,, [1]). For the latter definition we also requireX to preserve distributivity.

Definition 3.4. Let C be a small category and Y : C Top a functor. Denote by C Ythe category of elements of Y , that is, objects are pairs (c, x) consisting of an object c of Cand a point x Y (c). A morphism in C Y from (c, x) to (c, x) is a morphism : c c

in C satisfying the equation Y ()(x) = x.

Given Y : C Top, with the notation above, if we consider C Y as a topologicalcategory whose space of objects is

cC

Y (c)

then we have that the homotopy colimit of Y is the classifying space

B(C Y ) = hocolimCY,

that is, the realization of the nerve of the category C Y .Let X denote a commutative I-monoid. The category of elements IX is a permutative

category, that is, a symmetric monoidal category where associativity holds on the nose.According to [9], the classifying space of a permutative category has an E-space structure,and so we get that

hocolimIX

has an E-space structure. Here we think of an E-space as a space with an operation thatis associative and commutative up to a system of coherent homotopies. IfX is a commutativeI-rig, then I X is a bipermutative category and its classifying space is then an E-ringspace, that is, an E space with an operation that is associative and commutative (up

16 B. Villarreal

to coherent homotopy) that is distributive (up to coherent homotopy) over the E-spaceoperation.

Under certain assumptions of X there are homotopy equivalences

hocolimIX hocolimNX colimn

X [n]

where N denotes the subcategory of I with same set of objects and as arrows the canonicalinclusions. The right hand side equivalence requires the structural maps jn : X [n] X [n+1]to be cofibrations. The left hand side equivalence needs more background. Consider thesubcategory of I consisting of the same set of objects and all isomorphisms. We denote itas P. The (bi) permutative structure on I X restricts to P X , so that hocolimPX isalso an E space (E-ring space). In particular its group completion B(hocolimPX) is aninfinite loop space (E-ring space). The maps X [n] induce a map of (bi) permutativecategories P X P and therefore a map of infinite loop spaces (E-ring spaces)

X : B(hocolimPX) B(hocolimP).

It follows that the homotopy fiber hofib X is an infinite loop space (E-ring space). DenoteX := hocolimNX and X

+ its Quillen plus construction applied with respect to the maximal

perfect subgroup of 1(X). The following proposition is proved in [4].

Proposition 3.5. Suppose X is a commutative I-monoid (I-rig). Assume that X [n] is pathconnected and that the action of n on X [n] is homologically trivial for every n 0. Assumefurther that the commutator subgroup of 1(X) is perfect and X

+ is abelian. Then there is

a homotopy equivalence X+ hofib X .

The inclusion P I induces a map X : B(hocolimPX) B(hocolimIX). WhenX [n]is path connected for all n 0, hocolimIX is group like and hence we have the followingcommutative diagram

hofib X //

g

B(hocolimPX)X //

X

B(hocolimP)

hofib X1 // hocolimIX

X1 // hocolimI

Since hocolimI , we can identify hofib X1 with hocolimIX . The map g is a map of infinite

loop spaces, and in [4] they prove this is a homotopy equivalence under the conditions ofProposition 3.5, and thus hocolimIX X

+. Finally, for a commutative I-monoid (I-rig),

satisfying X [n] path connected with homologically trivial action of the symmetric group forall n 0 and X abelian, we conclude

X hocolimIX.


3.3 E-ring space structure of B(L,U)

Our first example and application of the machinery described in the previous section isshowing that {U(m)}m1 has a structure of a commutative I-rig U( ) which satisfies theconditions necessary to prove the classical result

colimm

U(m) = U

is an E-ring space. This will allow us to prove E-ring space structures on our spaces ofinterest.

Recall that m U(m) as permutation matrices, so that U(m) has a m action. Considerthe inclusions im : U(m) U(m+ 1)

A 7

(A 00 1

)

which are continuous and preserve group structure. The maps im restrict to the canonicalinclusions m m+1, therefore

im( A) = im()im(A)im()1 = im(A),

where m, A U(m) and on the right hand side m+1. Now let , r, m < r

and suppose both restrictions to the subset {1, ..., m} determine equal permutations in m.Denote i = ir ir1 im. Then, for A U(m),

i(A) =

((|m)A(|

1m ) 0

0 Irm

)=

((|m)A(

|1m ) 00 Irm

)= i(A).

Therefore U( ) : I Top is a functor. This I-space has a commutative I-rig structure asfollows. Let m,n : U(m) U(n) U(m + n) denote the block sum of matrices, whichis a group homomorphism. The (m,n) shue map U(n + m) U(n + m) is given byA 7 m,n A. We have the commutative diagram

U(m) U(n)

m,n // U(m+ n)

m,n

U(n) U(m)

n,m // U(m+ n)

where (A,B) = (B,A). Therefore U( ) is a commutative I-monoid. The other monoidalstructure is given by m,n : U(m)U(n) U(mn) the tensor product of matrices. Indeed,by definition m,n m,n(A,B) = n,m(A,B), where A U(m) and B U(n). Sincethe image m,n(U(m) U(n)) correspond to direct sum, then associativity, left and rightdistributivity over m,n hold.

Now we check the conditions of 3.5: the action of m on U(m) is homologically trivialsince conjugation action on U(m) is trivial up to homotopy, being U(m) path connected.The inclusions im are cellular and hence U( ) U . Finally, U is an H-space and thusabelian.

18 B. Villarreal

Proposition 3.6. Let L be a finitely generated cosimplicial group, then B(L, U( )) is acommutative I-rig.

Proof. Consider the I-rig U( ). Both the structural maps im and the action by elements ofm are continuous group homomorphisms and hence B(L, U( )) = B(L, )U( ) is an I-space.Also, block sum of matrices and tensor product are topological group morphisms so thatwith Lemma 3.2 we can define

m,n = B(L,m,n)(B(L, p1)B(L, p2))1 and m,n = B(L,m,n)(B(L, p1)B(L, p2))

1,

where p1 : U(m) U(n) U(m) and p2 : U(m) U(n) U(n) are the projections. Letp1 : U(n) U(m) U(n) and p

2 : U(n) U(m) U(m) denote also projections. Notice

that B(L, p2)B(L, p

1) = B(L, p1)B(L, p2) B(L, )

(where as before, is the symmetry morphism in Top). This implies that all propertiessatisfied by m,n and m,n will be preserved by m,n and m,n.

Theorem 3.7. Let L be finitely generated cosimplicial group and suppose that the spaceHom(L0, U(m)) is path connected for all m 1. Then, B(L, U) is an E-ring space.

Proof. By Proposition 3.6, hocolimIB(L, U( )) is an E-ring space. To show that this ho-motopy colimit is homotopy equivalent to B(L, U), first, note that the conjugation actionof n is homologically trivial since it factors through conjugation action on U(m). Using[8, Theorem 11.12] with our hypothesis, |Hom(L, U(m))| = B(L, U(m)) is path connectedfor all m 1. The colimit B(L, U) is also an H-space under block sum of matrices, andtherefore abelian.

Example 3.8. The property 0(Hom(L0, U(m))) = 0 for all m 1, is satisfied by thecosimplicial groups L = F/q, F/F (q), F/q, and F/F (q) defined in Example 2.13 and alsofor the surface groups T+1.

Remark 3.9. The results in this section also apply for the groups SU and Sp. For SO andO the proofs are not exactly similar, but still true. The arguments used in [4, Theorem 4.1and 4.2] also apply in our case.

References

[1] A. Adem, A. Bahri, M. Bendersky, F. Cohen, and S. Gitler. On decomposing suspensions ofsimplicial spaces, Boletin Sociedad Matematica Mexicana (3) Vol.15 (2009), 91-102

[2] A. Adem, F. Cohen, J. Gomez. Commuting elements, simplicial spaces and filtrations ofclassifying spaces, Math. Proc. Cambridge Philos. Soc. Vol. 152 (2012), 1, 91-114.

[3] A. Adem, F. Cohen, J. Gomez. Stable splittings, spaces of representations and almost com-muting elements in Lie groups, Math. Proc. Camb. Phil. Soc. Vol. 149 (2010), 455-490.

References 19

[4] A. Adem, J. Gomez, J. Lind, U. Tillman. Infinite loop spaces and nilpotent K-theory,arXiv:1503.02526 preprint 2014.

[5] O. Antolin, B. Villarreal, in preparation.

[6] H. Hironaka. Triangulations of algebraic varieties, Proc. Sympos. Pure Math. 29 (1975)165-185.

[7] T. Matumoto, M. Shiota. Unique triangulation of the orbit space of a differential transfor-mation group and its applications, Adv. Stud. Pure Math. 9 (1986), 41?55.

[8] J. P. May. The geometry of iterated loop spaces, Springer Lecture notes in mathematics,Springer, Berlin (1972).

[9] J. P. May. E-spaces, group completions and permutative categories, In new developmentsin topology (Proc. Sympos. Algebraic Topology, Oxford, 1972) pages 61-93. London Math.Soc. Lecture Notes Ser., No11 Cambridge Univ. Press, London, 1974.

[10] A. L. Onischnick, E. B. Vinberg. Lie Groups and Algebraic Groups, Translated from theRussian by D. A. Leites, Springer-Verlag, 1990.

[11] R. S. Palais. The Classification of G-spaces, Mem. Amer. Math. Soc. 36 (1960).

[12] D.H. Park and D.Y. Suh. Equivariant semi-algebraic triangulation of real algebraic G-varieties, Kyushu J. Math. 50(1) (1996), 179-205.

Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2,

Canada

E-mail adress: [email protected]

1 Introduction2 Homotopy Stable Decompositions2.1 Spaces of Homomorphisms2.2 Triangulation of Semi-algebraic Sets2.3 Simplicial Spaces and Homotopy Stable Decompositions2.4 Homotopy Stable Decomposition of Hom(Ln,G)2.5 Equivariant Homotopy Stable Decomposition of Hom(Fn/Kn,G)

3 B(L,U) as an E-ring space3.1 The Bar Construction of Hom(L,G)3.2 Relation between commutative I-monoids and infinite loop spaces3.3 E-ring space structure of B(L,U)

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