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Spaces of homomorphisms and group cohomology Torres Giese, Enrique 2007

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SPACES OF HOMOMORPHISMS AND GROUP COHOMOLOGY by Enrique Torres Giese A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA December 2007 c© Enrique Torres Giese, 2007 ii Abstract In this work we study the space of group homomorphisms Hom(Γ, G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups. Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Commuting Elements in Lie Groups . . . . . . . . . . . . . . . . . . 7 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The spaces Hom(Zn, SO(3)) . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Fundamental group of Hom(Zn, G) . . . . . . . . . . . . . . . . . . . . . 12 2.4 Homological computations . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Rational homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6 Noncommuting elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6.1 The case of SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6.2 The case of U(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1 Filtrations of BG with spaces of homomorphisms . . . . . . . . . . . . . 42 3.2 B(2, G) for Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 B(q,G) and group cohomology . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 The homotopy of B(q,G) . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.5 The homology of B(q,G) . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 iii iv Acknowledgements My deepest gratitude is to God who has shown sincere love for me and who has been taking care of me and my family during these years of study. I am also thankful to Him for the family that he has given to me, I have been blessed with a wise woman and with an adorable daughter. They have been a tremendous support and have brought lots of joy to my heart. I love you Maribel and Sarai!! Since I started studying Topology in México I conjectured that the word “Topology” was homotopic to the word “Generosity”, and my adviser, Alejandro Ádem, has been proving that to me since I met him in Madison. I sincerely acknowledge his support and patience with me. I also acknowledge the generous support of the National Council for Science and Technology (Conacyt-México). I am undoubtedly in debt with México. Finally, I would like to thank the Mathematics Department at UBC as well as the Pacific Institute for the Mathematical Sciences, for the excellent environment that they provide to work. After these years I am convinced that UBC was the best option for me to continue my studies. 1Chapter 1 Introduction In this work we will study the spaces of group homomorphisms from a geometric point of view and from a simplicial point of view as well. The space of group homomorphisms from Γ into G is denoted by Hom(Γ, G), and it arises in different contexts (see [16]) as this encodes rich geometric information. The first natural piece of information about the space of group homomorphisms is its number of components. This has been studied in Differential Geometry when the source group Γ is the fundamental group of a manifold M and G is a Lie group, as this space may be used to construct G-bundles over M . Since conjugate representations give rise to isomorphic G-bundles it is natural to consider the space of homorphisms modulo the action of G by conjugation on the space. A basic fact is that when G is connected, the induced map pi0(Hom(Γ, G)) → pi0(Hom(Γ, G)/G) is a bijection of sets. In this regard W. Goldman proved the following Theorem 1.1 ([12]) If Mg is a closed orientable surface of genus g ≥ 1, then 1. #pi0(Hom(pi1(Mg), SL(2,R))) = 2 2g+1 + 2g − 3, 2. #pi0(Hom(pi1(Mg), G)) = 2 if G is SO(3) or PSL(2,C), 3. #pi0(Hom(pi1(Mg), G)) = 1 if G is SU(2) or SL(2,C), and J.Li using Complex Geometry proved Theorem 1.2 ([19]) Let G be a connected complex semi-simple Lie group and let Mg be a closed oriented surface of genus g > 1, 21. then #pi0(Hom(pi1(Mg), G)) = #pi1(G), 2. and if G is also simply connected, then pi1(Hom(pi1(Mg), G)) = 1. More applications of the space of homomorphisms occur, for instance, in Physics. The orbit space Hom(Zn, G)/G, with G acting by conjugation, is the moduli space of isomorphism classes of flat connections on principal G-bundles over the n-dimensional torus. Questions concerning the understanding of the structure of the components of this moduli space are part of the study of the quantum field theory of gauge theories over the n-dimensional torus (see [7],[18]). More recently, in Algebraic Toplogy, Adem and Cohen in [1] studied the homotopy of the space Hom(Zn, G), which is just the space of ordered commuting n-tuples of elements from G, topologized as a subspace of Gn. They show that Hom(Zn, G) is connected if the maximal abelian subgroups of G are connected; find a stable splitting of Hom(Zn, G) when G is a closed subgroup of GLn(C); and compute the integral homology of Hom(Z3, SU(2)). In this work we also study these spaces from the point of view of Algebraic Topology by computing classic invariants of them. In Chapter 2 we will focus on the structure of commuting n-tuples in a Lie group G. We study the space Hom(Zn, G), where G is one of SO(3), SU(2) or U(2). We calculate the fundamental groups of the connected components of these spaces by showing that if Hom(Zn, G)+ is the component of the trivial representation, then Theorem 1.3 The inclusion maps Hom(Zn, G)+ → G n if G = SO(3), SO(4) Hom(Zn, G)→ Gn if G = SU(2), U(2) are isomorphisms on pi1 for all n ≥ 1. 3When G = SO(3) the space of commuting n-tuples in G is not connected. We com- pute the number of components of this space and we also identify each of its components. Theorem 1.4 #pi0(Hom(Z n, SO(3))) = 1 +   1 6 (4n − 3× 2n + 2) if n is even 2 3 (4n−1 − 1)− 2n−1 + 1 if n is odd One component is Hom(Zn, SO(3))+, and all others are homeomorphic to S 3/Q8, where Q8 is the quaternion group of eight elements. We also obtain the mod-2 and rational cohomology of the components of the space Hom(Zn, SO(3)) and its multiplicative structure when n = 2, 3. For instance we obtain Theorem 1.5 The Poincaré polynomial of Hom(Zn, SO(3))+ is given by 1 + ( n 2 ) t2 + ( n 1 ) t3 + ( n 4 ) t4 + ( n 3 ) t5 + · · ·+ ( n n− 3 ) tn−1 + tn + ( n n− 1 ) tn+1 when n is even, and 1 + ( n 2 ) t2 + ( n 1 ) t3 + ( n 4 ) t4 + ( n 3 ) t5 + · · ·+ ( n n− 1 ) tn−1 + ( n n− 2 ) tn + tn+2 when n is odd. At the end of Chapter 2 we will consider the dual of Hom(Zn, G), that is, the space of noncommuting n-tuples in G. We will study these spaces when G is SO(3) or U(2). Our simplicial approach is developed in Chapter 3. In this chapter we use the spaces of homomorphisms to define a filtration of the classifying space of any topological group. The strategy is to let the source group vary over a family of groups to define a family of simplicial subspaces, in a natural way, of the bar-construction of G. The first stage of this filtration is determined by the structure of the commuting elements in G, the second by the elements in G defining a subgroup of nilpotency class < 3, etc. The 4precise definition of these spaces is as follows. Let q ≥ 2 fixed and let Fn be the free group of rank n. We denote by Γqn the q th-stage of the lower central series of Fn, that is Γqn = [Fn,Γ q−1 n ]. Definition 1.6 Let q ≥ 2 and G be a topological group. 1. The space B(q,G) is the geometric realization of the simplicial space with n- simplices given by Bn(q,G) = Hom(Fn/Γ q n, G). 2. The space E(q,G) is the geometric realization of the simplicial space with n- simplices given by Bn(q,G) = G×Hom(Fn/Γ q n, G). It turns out that these simplicial spaces are subspaces of the usual simplicial spaces that define BG and EG respectively. Some of the main properties of these spaces are contained in the following Theorem 1.7 Let G be a topological group. 1. There are natural morphisms of principal G-bundles E(2, G) //  · · · // E(q,G) //  E(q + 1, G)  // · · · // EG  B(2, G) // · · · // B(q,G) // B(q + 1, G) // · · · // BG 2. The horizontal maps are cofibrations and yield a natural filtration of EG and BG. 3. The homotopy fiber of B(q,G)→ BG is E(q,G). 4. If G is finite and q ≥ 2, then Hi(B(q,G),Z) is finite for all i > 0 and has torsion only at primes dividing the order of G. The stucture of the spaces B(q,G) depends on the structure of the lattice of sub- groups of G of nilpotency class < q. This means that when q = 2 we need to consider 5the commuting elements in G to describe the space B(2, G). Is in this case in which we use some results from Section 2.5 to study the space B(2, G) when G is a Lie group. On the other hand, if G is discrete, there is a family of finite groups for which a description of B(2, G) can be made explicitly, namely, the transitively commutative groups (TC). These groups are those whose centralizers of noncentral elements are abelian. The following two results describe the space B(2, G) when G is a TC group. Theorem 1.8 If G is a TC group, then 1. B(2, G) is a K(pi, 1), with pi isomorphic to the free product of the maximal abelian subgroups of G amalgamated along the center of G 2. E(2, G) is a K(pi, 1), with pi a free group of rank 1− |G : Z(G)|+ k∑ i=1 (|G : Z(G)| − |G : Mi|) where M1, . . . ,Mk are the maximal abelian subgroups of G. Corollary 1.9 If G is a TC group with trivial center and M1, . . . ,Mk are the distinct maximal abelian subgroups of G, then B(2, G) ' ∨ 1≤i≤k  ∏ p||Mi| BP   where P ∈ Sylp(G) and np,i is the number of Sylow p-subgroups of G contained in Mi. In particular, H1(B(2, G)) = ⊕ p||G| P np where P ∈ Sylp(G) and np is the number of Sylow p-subgroups of G. A class similar to TC groups was studied by Suzuki (see [23]) to get a result that paved the way to the celebrated Feit-Thompson theorem. This class is called CA, “cen- tralizer abelian”, and it is characterized by the following propery: if 1 6= x ∈ G, then 6CG(x) is abelian. Suzuki’s result asserts that a CA group of odd order has to be a Frobenius group or an abelian group, and hence solvable. In fact, it turns out that the spaces B(q,G) are related to the Feit-Thompson theorem as the following turn out to be equivalent: A. (Feit-Thompson) Every finite group of odd order is solvable. B. If G has odd order, then H1(E(2, G);Z)→ H1(B(2, G);Z) is not surjective. Finally, we close this chapter by discussing the homology and homotopy of the space B(q,G) for any finite group G. 7Chapter 2 Commuting Elements in Lie Groups 2.1 Introduction In this chapter we will calculate the fundamental groups of the connected components of the spaces Hom(Zn, G), where G is one of SO(3), SU(2) or U(2), and we will also obtain the mod-2 and rational cohomology of the components of Hom(Zn, SO(3)). The space Hom(Zn, G) is just the space of ordered commuting n- tuples of elements from G, topologized as a subset of Gn. The spaces Hom(Zn, SU(2)), Hom(Zn, U(2)) are connected (see [1]), but the spaceHom(Zn, SO(3)) has many compo- nents if n > 1. One of the components is the one containing the element (id, id, . . . , id); see Section 2.2. The other components are all homeomorphic to V2(R 3)/Z2 ⊕ Z2, where V2(R 3) is the Stiefel manifold of orthonormal 2-frames in R3 and the action of Z2 ⊕ Z2 on V2(R 3) is given by (1, 2)(v1, v2) = (1v1, 2v2), where j = ±1 and (v1, v2) ∈ V2(R 3). Let e1, e2, e3 be the standard basis of R 3. Under the homeomorphism SO(3) → V2(R 3) given by A 7→ (Ae1, Ae2) the action of Z2⊕Z2 on V2(R 3) corresponds to the action defined by right multiplication by the elements of the group generated by the transformations (x1, x2, x3) 7→ (x1,−x2,−x3), (x1, x2, x3) 7→ (−x1, x2,−x3). The orbit space of this action is homeomorphic to the 3-manifold S3/Q8, where Q8 is the quaternion group of order eight. 8Then Hom(Zn, SO(3)) will be a disjoint union of many copies of S3/Q8 and the component containing (id, . . . , id). For brevity let ~1 denote the n-tuple (id, . . . , id). Definition 2.1 Let Hom(Zn, SO(3))+ be the component of the trivial representation ~1 in Hom(Zn, SO(3)), and let Hom(Zn, SO(3))− be the complement Hom(Z n, SO(3)) \ Hom(Zn, SO(3))+. Our main result, to be proved in Section 2.3, is the following Theorem 2.2 For all n ≥ 1 pi1(Hom(Z n, SO(3))+) = Z n 2 pi1(Hom(Z n, SU(2))) = 0 pi1(Hom(Z n, U(2))) = Zn The other components of Hom(Zn, SO(3)), n > 1, all have fundamental group the quaternion group Q8. Remark 2.3 To prove this theorem we first prove that pi1(Hom(Z n, SO(3))+) = Z n 2 , and then use the following property of spaces of homomorphisms (see [12]). Let Γ be a discrete group, p : G̃ → G a covering of Lie groups, and C a component of the image of the induced map p∗ : Hom(Γ, G̃) → Hom(Γ, G). Then p∗ : p −1 ∗ (C) → C is a regular covering with covering group Hom(Γ,Ker p). Applying this to the universal coverings SU(2)→ SO(3) and SU(2)× R→ U(2) we get the following coverings Zn2 → Hom(Z n, SU(2))→ Hom(Zn, SO(3))+ Zn → Hom(Zn, SU(2))× Rn → Hom(Zn, U(2)) The computation of the fundamental groups in Theorem 2.2 as well as the corre- sponding homological computations in Section 2.4 were obtained in [25]. 92.2 The spaces Hom(Zn, SO(3)) In this section we describe the topology of the spaces Hom(Zn, SO(3)), n ≥ 2. If A1, A2 are commuting elements from SO(3) then there are 2 possibilities: either A1, A2 are rotations about a common axis; or A1, A2 are involutions about axes meeting at right angles. The first possibility covers the case where one of A1, A2 is the identity since the identity can be considered as a rotation about any axis. Thus there are two possibilities for an n-tuple (A1, . . . , An) ∈ Hom(Z n, SO(3)): 1. Either A1, . . . , An are all rotations about a common axis L; or 2. There exists at least one pair i, j such that Ai, Aj are involutions about perpen- dicular axes. If vi, vj are unit vectors representing these axes then all the other Ak must be one of id, Ai, Aj or AiAj = AjAi (the involution about the cross product vi × vj). It is clear that if ω(t) = (A1(t), . . . , An(t)) is a path in Hom(Z n, SO(3)), then exactly one of the following two possibilities has to occur: either the rotations A1(t), . . . , An(t) have a common axis L(t) for all t; or there exists a pair i, j such that Ai(t), Aj(t) are involutions about perpendicular axes for all t. In the second case the pair i, j does not depend on t. Proposition 2.4 Hom(Zn, SO(3))+ is the space of n-tuples (A1, . . . , An) in SO(3) n such that all the Aj have a common axis of rotation. Proof. Let A1, . . . , An have a common axis of rotation L. Thus A1, . . . , An are rotations about L by some angles θ1, . . . , θn. We can change all angles to 0 by a path (while maintaining the common axis). Conversely, if ω(t) = (A1(t), . . . , An(t)) is a path containing ~1 then the Aj(t) will have a common axis of rotation for all t (which might change with t).  Thus any component ofHom(Zn, SO(3))− can be represented by an n-tuple (A1, . . . , An) satisfying possibility 2 above. 10 Corollary 2.5 The components of Hom(Z2, SO(3)) are Hom±(Z 2, SO(3)). Proof. Let (A1, A2) be a pair in Hom(Z 2, SO(3))−. Then there are unit vectors v1, v2 in R3 such that v1, v2 are perpendicular and A1, A2 are involutions about v1, v2 respec- tively. The pair (v1, v2) is not unique since any one of the four pairs (±v1,±v2) will deter- mine the same involutions. In fact there is a 1-1 correspondence between pairs (A1, A2) in Hom(Z2, SO(3))− and sets {(±v1,±v2)}. Thus Hom(Z 2, SO(3))− is homeomorphic to the orbit space V2(R 3)/Z2⊕Z2. Since V2(R 3) is connected so is Hom(Z2, SO(3))−.  Next we determine the number of components of Hom(Zn, SO(3))− for n > 2. The following example will give an indication of the complexity. Example 2.6 Let (A1, A2, A3) be an element of Hom(Z 3, SO(3))−. Then there exists at least one pair Ai, Aj without a common axis of rotation. For example suppose A1, A2 don’t have a common axis. Then A1, A2 are involutions about perpendicular axes v1, v2, and there are 4 possibilities for A3 : A3 = id, A1, A2 or A3 = A1A2. We will see that the triples (A1, A2, id), (A1, A2, A1), (A1, A2, A2), (A1, A2, A1A2) belong to different components. Similarly if A1, A3 or A2, A3 don’t have a common axis of rotation. This leads to 12 possible components, but some of them are in fact the same component. An analysis leads to a total of 7 distinct components corresponding to the following 7 triples: (A1, A2, id), (A1, A2, A1), (A1, A2, A2), (A1, A2, A1A2), (A1, id, A3), (A1, A1, A3), (id, A2, A3); where A1, A2 are distinct involutions in the first 4 cases; A1, A3 are distinct involutions in the next 2 cases; and A2, A3 are distinct involutions in the last case. These components are all homeomorphic to S3/Q8. Thus Hom(Z 3, SO(3)) is homeomorphic to the disjoint union Hom(Z3, SO(3))+ unionsq S 3/Q8 unionsq . . . unionsq S 3/Q8, where there are 7 copies of S3/Q8. 11 The pattern of this example holds for all n ≥ 3. A simple analysis shows that the space Hom(Zn, SO(3))− consists of n-tuples ~A = (A1, . . . , An) ∈ SO(3) n satisfying the following conditions: 1. Each Ai is either an involution about some axis vi, or the identity. 2. If Ai, Aj are distinct involutions then their axes are at right angles. 3. There exists at least one pair Ai, Aj of distinct involutions. 4. If Ai, Aj are distinct involutions then every other Ak is one of id, Ai, Aj or AiAj. This leads to 5 possibilities for any element (B1, . . . , Bn) ∈ Hom(Z n, SO(3))− : (B1, B2, ∗, . . . , ∗), (B1, id, ∗, . . . , ∗), (id, B2, ∗, . . . , ∗), (B1, B1, ∗, . . . , ∗), (id, id, ∗, . . . , ∗), where B1, B2 are distinct involutions about perpendicular axes and the asterisks are choices from amongst id, B1, B2, B3 = B1B2. The choices must satisfy the conditions above. These 5 cases account for all components of Hom(Zn, SO(3))−, but not all choices lead to distinct components. If ω(t) = (B1(t), B2(t), . . . , Bn(t)) is a path in the space Hom(Zn, SO(3))− then it is easy to verify the following statements: 1. If some Bi(0) = id then Bi(t) = id for all t. 2. If Bi(0) = Bj(0) then Bi(t) = Bj(t) for all t. 3. If Bi(0), Bj(0) are distinct involutions then so are Bi(t), Bj(t) for all t. 4. If Bk(0) = Bi(0)Bj(0) then Bk(t) = Bi(t)Bj(t) for all t. These 4 statements are used repeatedly in the proof of the next theorem. 12 Theorem 2.7 The number of components of Hom(Zn, SO(3))− is  1 6 (4n − 3× 2n + 2) if n is even 2 3 (4n−1 − 1)− 2n−1 + 1 if n is odd Moreover, each component is homeomorphic to S3/Q8. Proof. Let xn denote the number of components. The first 3 values of xn are x1 = 0, x2 = 1 and x3 = 7, in agreement with the statement in the theorem. We consider the above 5 possibilities one by one. First assume ~B = (B1, B2, ∗, . . . , ∗). Then different choices of the asterisks lead to different components. Thus the contribution in this case is 4n−2. Next assume ~B = (B1, id, ∗, . . . , ∗). Then all choices for the asterisks are admissible, except for those choices involving only id and B1. This leads to 4 n−2 − 2n−2 possibilities. However, changing every occurrence of B2 to B3, and B3 to B2, keeps us in the same component. Thus the total contribution in this case is (4n−2− 2n−2)/2. This is the same contribution for cases 3 and 4. Finally, there are xn−2 components associated to ~B = (id, id, ∗, . . . , ∗). This leads to the recurrence relation xn = 4 n−2 + 3 2 (4n−2 − 2n−2) + xn−2 Solving this recurrence relation for the xn leads to the desired formula. Given any element (B1, . . . , Bn) ∈ Hom−(Z n, SO(n)) we select a pair of involutions, say Bi, Bj , with perpendicular axes vi, vj. All the other Bk are determined uniquely by Bi, Bj. Thus the element (vi, vj) ∈ V2(R 3) determines (B1, . . . , Bn). But all the elements (±vi,±vj) also determine (B1, . . . , Bn). Thus the component to which (B1, . . . , Bn) be- longs is homeomorphic to V2(R 3)/Z2 ⊕ Z2 ∼= S 3/Q8.  2.3 Fundamental group of Hom(Zn, G) In this section we prove Theorem 2.2, and we start by finding an appropriate description of Hom(Zn, SO(3))+. Let T n = (S1)n denote the n-torus. Then 13 Theorem 2.8 The space Hom(Zn, SO(3))+ is homeomorphic to the quotient space S 2× T n/ ∼, where ∼ is the equivalence relation generated by (v, z1, . . . , zn) ∼ (−v, z̄1, . . . , z̄n) and (v,~1) ∼ (v ′,~1) for all v, v′ ∈ S2, zi ∈ S 1. Proof. If (A1, . . . , An) ∈ Hom(Z n, SO(3))+ then there exists v ∈ S 2 such that A1, . . . , An are rotations about v. Let zj ∈ S 1 be the elements corresponding to these rotations. The (n+1)-tuple (v, z1, . . . , zn) is not unique. For example, if one of the Ai’s is not the identity then (−v, z̄1, . . . , z̄n) determines the same n-tuple of rotations. On the other hand, if all the Ai’s are the identity then any element v ∈ S 2 is an axis of rotation.  We will use the notation [v, z1, . . . , zn] to denote the equivalence class of the (n+1)- tuple (v, z1, . . . , zn). Thus x0 = [v,~1] ∈ S 2×T n/ ∼ is a single point, which we choose to be the base point. It corresponds to the n-tuple (id, . . . , id) ∈ Hom(Zn, SO(3))+. Then for n > 1 the space Hom(Zn, SO(3))+ is locally homeomorphic to R n+2 everywhere except at the point x0 where it is singular (this space is not a manifold, see Proposition 2.11). Proof of Theorem 2.2. Notice that the result holds for n = 1 since Hom(Z, G) is homeomorphic to G. The first step is to compute pi1(Hom(Z n, SO(3))+). Let T n 0 = T n − {~1} and Hn+ = Hom(Z n, SO(3))+. Removing the singular point x0 = [v,~1] from Hn+ we have H n + − {x0} ∼= S2 × T n0 /Z2, see Theorem 2.8. If t denotes the generator of Z2 then the Z2 action on S 2 × T n0 is given by t(v, z1, . . . , zn) = (−v, z̄1, . . . , z̄n), v ∈ S 2, zj ∈ S 1 This action is fixed point free and so there is a two-fold covering S2 × T n0 p → Hn+ − {x0} and a short exact sequence 1→ pi1(S 2 × T n0 )→ pi1(H n + − {x0})→ Z2 → 1 Let n denote the north pole of S2. Then for base points in S2 × T n0 and H n + − {x0} we take (n,−1, . . . ,−1) = (n,−~1) and [n,−1, . . . ,−1] = [n,−~1] respectively. 14 This sequence splits. To see this note that the composite S2 → S2× T n0 → S 2 is the identity, where the first map is v 7→ (v,−~1) and the second is just the projection. Both maps are equivariant with respect to the Z2-actions, and therefore H n + − {x0} retracts onto RP 2. First we consider the case n = 2. Choose −1 to be the base point in S1. The above formula for the action of Z2 also defines a Z2 action on S 2 × (S1 ∨ S1). This action is fixed point free. The inclusion S2 × (S1 ∨ S1) ⊂ S2 × S1 × S1 is equivariant and there exists a Z2-equivariant strong deformation retraction from S 2× T 20 onto S 2× (S1 ∨ S1). Let a1, a2 be the generators (n, S 1,−1) and (n,−1, S1) of pi1(S 2 × T 20 ) = Z ∗ Z. The involution t : S2 × T 20 → S 2 × T 20 induces isomorphisms pi1(S 2 × (S1 ∨ S1), {n,−1,−1}) c → pi1(S 2 × (S1 ∨ S1), {s,−1,−1}) pi1(S 2 ∨ (S1 ∨ S1), {n,−1,−1}) c → pi1(S 2 ∨ (S1 ∨ S1), {n,−1,−1}) where s = −n is the south pole in S2. We have the following commutative diagram S2 ∨n (S 1 ∨ S1) in  t // S2 ∨s (S 1 ∨ S1) is  S2 ×n (S 1 ∨ S1) t // p ((QQ QQ QQ QQ QQ QQ S2 ×s (S 1 ∨ S1) p vvmmm mm mm mm mm m H2+ − {x0} where in and is are inclusions. Here the subscripts n and s refer to the north and south poles respectively, which we take to be base points of S2 in the one point unions. The inclusions in, is induce isomorphisms on pi1 and therefore we get p∗pi1(S 2∨n (S 1∨S1)) = p∗pi1(S 2∨s (S 1∨S1)). Let a1 and a2 be the canonical generators of the fundamental group of S1 ∨ S1. Thus t sends a1 to the loop based at s but with the opposite orientation (similarly for a2). We now have pi1(H 2 + − {x0}) = 〈a1, a2, t | t 2 = 1, at1 = a −1 1 , a t 2 = a −1 2 〉. 15 For the computation of pi1(H n + − {x0}), n ≥ 3, note that the inclusion T n 0 ⊂ T n induces an isomorphism on pi1. Thus pi1(T n 0 ) = 〈a1, . . . , an | [ai, aj ] = 1 ∀ i, j〉. The various inclusions of T 20 into T n 0 (corresponding to pairs of generators) show that the action of t on the generators is still given by ati = a −1 i . Thus pi1(H n + − {x0}) = 〈a1, . . . , an, t | t 2 = 1, [ai, aj ] = 1, a t i = a −1 i 〉, for n ≥ 3. The final step in the calculation of pi1(H n +) is to use van Kampen’s theorem. To do this let U ⊂ S1 be a small open connected neighbourhood of 1 ∈ S1 which is invariant under conjugation. Here small means −1 6∈ U . Then Nn = S 2×Un/ ∼ is a contractible neighbourhood of x0 in H n +. We apply van Kampen’s theorem to the situation H n + = (Hn+ − {x0}) ∪Nn. The intersection Nn ∩ (H n +−{x0}) is homotopy equivalent to (S 2× Sn−1)/Z2 where Z2 acts by multiplication by −1 on both factors. Therefore pi1(Nn ∩ (H n + − {x0})) ∼= Z when n = 2, and Z2 when n ≥ 3. Thus we need to understand the homomorphism induced by the inclusion Nn ∩ (H n + − {x0})→ H n + − {x0}. When n = 2 the inclusion of N2 ∩ (H 2 +−{x0}) into H 2 +−{x0} induces the following commutative diagram Z // 2  Z ∗ Z  pi1(N2 ∩ (H 2 + − {x0})) //  pi1(H 2 + − {x0})  Z2 = // Z2 where the map on top is the commutator map. So if the generator of the group pi1(N2 ∩ (H2+ − {x0})) is sent to w ∈ pi1(H 2 + − {x0}), then w 2 = [a1, a2], and the image of w in Z2 is t. Thus we can write w = a n1 1 a m1 2 · · · a nr 1 a mr 2 t with ni,mi ∈ Z. Then w2 = an11 a m1 2 · · · a nr 1 a mr 2 a −n1 1 a −m1 2 · · · a −nr 1 a −mr 2 = a1a2a −1 1 a −1 2 16 which occurs only if r = 1 and n1 = m1 = 1. It follows that w = a1a2t. Thus pi1(H 2 +) = 〈a1, a2, t | t 2 = 1, at1 = a −1 1 , a t 2 = a −1 2 , a1a2t = 1〉 and routine computations show that this is the Klein four group. For n ≥ 3 the inclusion map Nn ∩ (H n + − {x0})→ H n + − {x0} can be understood by looking at the following diagram S2 × S1 //  ))TTT TTT TTT TTT TTT T S2 × T 20  ''OO OO OO OO OO O S2 × Sn−1 //  S2 × T n0  N2 ∩ (H 2 + − {x0}) // ))TTT TTT TTT TTT TTT H2+ − {x0} ''OO OO OO OO OO O Nn ∩ (H n + − {x0}) // H n + − {x0} Note that the map N2∩(H 2 +−{x0})→ Nn∩(H n +−{x0}) induces the canonical projection Z → Z2. A diagram chase argument shows that the inclusion Nn ∩ (H n + − {x0}) → Hn+ − {x0} imposes the relation a1a2t as well, and therefore pi1(H n +) = 〈a1, . . . , an, t | t 2 = 1, [ai, aj ] = 1, a t i = a −1 i , a1a2t = 1〉. As the ai’s commute the last two relations show that a 2 i = 1 for all i, whereas the last relation shows that t can be omitted from the list of generators. Thus we see that this group is isomorphic to Zn2 . This completes the proof of Theorem 2.2 for SO(3). The cases of SU(2) and U(2) follow from Remark 2.3.  Since the map pi1(∨nG) → pi1(G n) is an epimorphism, it follows that the inclusion maps Hom(Zn, G)+ → G n if G = SO(3) Hom(Zn, G)→ Gn if G = SU(2), U(2) 17 are isomorphisms in pi1 for all n ≥ 1. Recall that there is a map Hom(Γ, G) → Map∗(BΓ, BG), where Map∗(BΓ, BG) is the space of pointed maps from the classi- fying space of Γ into the classifying space of G. Let Map∗(T n, BG)0 be the component of the map induced by the trivial representation. Corollary 2.9 The maps Hom(Zn, G)+ →Map∗(T n, BG)0 if G = SO(3) Hom(Zn, G)→Map∗(T n, BG)0 if G = U(2) are injective in pi1 for all n ≥ 1. Proof. By induction on n, with the case n = 1 being trivial. Assume n > 1, and note that there is a commutative diagram Hom(Zn, SO(3))+ //  Map∗(Bpi1(T n), BSO(3))0  Hom(pi1(T n−1 ∨ S1), SO(3))+ //  Map∗(Bpi1(T n−1 ∨ S1), BSO(3))0  Hom(pi1(T n−1), SO(3))+ × SO(3) //Map∗(Bpi1(T n−1), BSO(3))0 × SO(3) in which the bottom map is injective in pi1 by inductive hypothesis, the lower vertical maps are homeomorphisms, and the upper left vertical map is injective in pi1. Thus the map on top is also injective as wanted. The proof for U(2) is the same.  Remark 2.10 We have the following observations. 1. The two-fold cover S3 × S3 → SO(4) allows us to study Hom(Zn, SO(4)). Let Hom(Zn, SO(4))+ be the component covered byHom(Z n, S3×S3). Since the space Hom(Zn, S3×S3) is homeomorphic to Hom(Zn, S3)×Hom(Zn, S3), it follows that pi1(Hom(Z n, SO(4))+) = Z n 2 18 2. The space Hom(Z2, SO(4)) has two components. One is Hom(Z2, SO(4))+, which is covered by ∂−1SU(2)2(1, 1), and the other is covered by ∂ −1 SU(2)2(−1,−1), where ∂SU(2)2 is the commutator map of SU(2) × SU(2). Recall ∂ −1 SU(2)(−1) is home- omorphic to SO(3) (see [4]), so ∂−1SU(2)2(−1,−1) is homeomorphic to SO(3) × SO(3)/Z2 × Z2, where the group Z2 × Z2 acts by left diagonal multiplication when it is thought of as the subgroup of SO(3) generated by the transformations (x1, x2, x3) 7→ (x1,−x2,−x3) and (x1, x2, x3) 7→ (−x1, x2,−x3). 3. Corollary 2.9 holds similarly for SO(4), and trivially for SU(2). 2.4 Homological computations In this section we compute the mod-2 and rational cohomology of Hom(Zn, SO(3))+, and we also discuss its multiplicative structure in the mod-2 case when n = 2, 3. We close this section by exploring the case of U(2). The mod-2 cohomology of the other components of Hom(Zn, SO(3)) is well-known since these are all homeomorphic to S3/Q8. To perform the computation we will use the description of Hom(Zn, SO(3))+ that we saw in the proof of Theorem 2.2. The ingredients we have to consider are the spectral sequence of the fibration S2 × T n0 → (Hn+ − {x0})→ RP ∞ whose E2-term is Z2[u]⊗ E(v)⊗ E(x1, . . . , xn)/(x1 · · · xn) with deg(u) = (1, 0), deg(v) = (0, 2) and deg(xi) = (0, 1); and the spectral sequence of the fibration S2 × Sn−1 → Nn ∩ (H n + − {x0})→ RP ∞ whose E2-term is Z2[u]⊗ E(v)⊗ E(w) with deg(u) = (1, 0), deg(v) = (0, 2) and deg(w) = (0, n − 1). Note that in both cases d2(v) = 0 and d3(v) = u 3, whereas d2(xi) = 0 sinceH 1(Hn+−{x0};Z2) = Z n+1 2 . Therefore the first spectral sequence collapses at the fourth term. As dn(w) = u n and dj(w) = 0 19 for j 6= n, the second spectral sequence collapses at the third term when n = 2 and at the fourth term when n ≥ 3. By naturality we can compute the homomorphisms in cohomology induced by the inclusion map Nn ∩ (H n + − {x0}) → H n + − {x0} from the E∞-terms of our spectral sequences, and thus we can compute the Mayer-Vietoris long exact sequence of the pair (Hn+ − {x0}, Nn). We obtain the following Theorem 2.11 For n = 2, 3, Hq(Hom(Z2, SO(3))+;Z2) =   Z2 q = 0 Z2 ⊕ Z2 q = 1 Z2 ⊕ Z2 ⊕ Z2 q = 2 Z2 ⊕ Z2 ⊕ Z2 q = 3 Z2 q = 4 0 q ≥ 5 Hq(Hom(Z3, SO(3))+;Z2) =   Z2 q = 0 Z2 ⊕ Z2 ⊕ Z2 q = 1 Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2 q = 2 Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2 q = 3 Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2 q = 4 Z2 q = 5 0 q ≥ 6 whereas for n ≥ 4, 20 Hq(Hom(Zn, SO(3))+;Z2) =   Z2 q = 0 Zn2 q = 1 Z (n1)+( n 2) 2 q = 2 Z ( nq−2)+( n q−1)+( n q) 2 3 ≤ q ≤ n Z ( nn−1)+1 2 q = n+ 1 Z2 q = n+ 2 0 q ≥ n+ 3 In particular, the Euler characteristic of Hom(Zn, SO(3))+ is zero for all n > 1. The rational cohomology of Hom(Zn, SO(3))+ can also be derived from our methods and these computations can be compared with those in [5] via the covering Hom(Zn, SU(2))→ Hom(Zn, SO(3))+ To obtain the rational cohomology one has to look at the 2-fold coverings S2 × T n0 → (Hn+ − {x0}) and S 2 × Sn−1 → Nn ∩ (H n + − {x0}) (see the proof of Theorem 2.2). We write H∗(S2 × Sn−1;Q) = Q[v, w]/(v2, w2) H∗(S2 × T n0 ;Q) = Q[v, x1, . . . , xn]/(v 2, x1 · · · xn) where deg(v) = 2, deg(w) = n− 1 and deg(xi) = 1. The action of Z2 on these spaces is given by τ · (v, w) = (−v, (−1)nw) and τ · (v, x1, . . . , xn) = (−v,−x1, . . . ,−xn). Notice 21 that H̄∗(S2 × Sn−1;Q)Z2 =   Qw if n evenQvw if n odd Thus the Mayer-Vietoris long exact sequence of the pair (Hn+ − {x0}, Nn) shows that 1. Hq(Hom(Zn, SO(3))+;Q) = H q(S2 × T n0 ;Q) Z2 for q ≤ n− 1. 2. When n is even Hn(Hom(Zn, SO(3))+;Q) = Q, Hn+1(Hom(Qn, SO(3))+;Q) = Q ( nn−1) and trivial for q ≥ n+ 2. 3. When n is odd Hn(Hom(Zn, SO(3))+;Q) = Q ( nn−2), Hn+1(Hom(Zn, SO(3))+;Q) = 0, Hn+2(Hom(Zn, SO(3))+;Q)) = Q and trivial for q ≥ n+ 3. Thus we obtain Theorem 2.12 The Poincaré polynomial of Hom(Zn, SO(3))+ is given by 1 + ( n 2 ) t2 + ( n 1 ) t3 + ( n 4 ) t4 + ( n 3 ) t5 + · · ·+ ( n n− 3 ) tn−1 + tn + ( n n− 1 ) tn+1 when n is even, and 1 + ( n 2 ) t2 + ( n 1 ) t3 + ( n 4 ) t4 + ( n 3 ) t5 + · · ·+ ( n n− 1 ) tn−1 + ( n n− 2 ) tn + tn+2 when n is odd. Remark 2.13 Note that the coverings used in the proof of the last two theorems show that there is no torsion at primes different from 2. 22 We now discuss the multiplicative structure of the mod-2 cohomology of the space Hom(Z2, SO(3))+. The integral cohomology of Hom(Z 2, SU(2)) is computed in [1]. This is done either by studying the cofibration S3 ∨ S3 → Hom(Z2, SU(2))→ S6 − SO(3) where S6 − SO(3) is the Spanier-Whitehead dual of SO(3) in S6; or by considering the complement of Hom(Z2, SU(2)) in SU(2) × SU(2), that is the non-commuting pairs, denoted by X2(SU(2)) (see Section 2.6), and the duality isomorphism Hq(SU(2) 2, X2(SU(2));Z) = H 6−q(Hom(Z2, SU(2));Z) The last isomorphism holds since the spaces Hom(Γ, G), with Γ a discrete group and G a Lie group, turn out to be analytic varieties and hence locally contractible (see [12] page 568, and [14]). The key ingredients are the equivalence X2(SU(2)) ' {(A,B) ∈ SU(2) 2 | [A,B] = −I} = SO(3), and the fact that {(A,B) ∈ SU(2)2 | [A,B] = −I} is actually a subspace of S2 × S2. The mod-2 case computation yields the following Hq(Hom(Z2, SU(2));Z2) =   Z2 q = 0 0 q = 1 Z2 q = 2 Z2 ⊕ Z2 ⊕ Z2 q = 3 Z2 q = 4 0 q ≥ 5 Note that in order to get the multiplicative structure it suffices to compute the square of the class of degree two. Let x be the class of degree two; a, b ∈ H3 the classes corresponding to S3 ∨S3; and y, z the classes of degree three and four respectively. The equivalence X2(SU(2)) ' SO(3) tells us that there is a deformation retraction from an open neighborhood in S3 × S3 onto this copy of SO(3), and hence a deformation 23 retraction from an open neighborhood V in S6 onto SO(3). Thus H∗(S6−SO(3);Z2) ∼= H∗(S6 − V ), and the action of Sq2 on H2(S6 − V ;F2) is trivial by a theorem of Thom ([22] page 30). Proposition 2.14 The mod-2 cohomology ring H∗(Hom(Z2, SU(2));F2) is an exterior algebra on x, y, z, a, b with trivial products. As for the mod-2 cohomology of Hom(Z2, SO(3))+ we get Proposition 2.15 The mod-2 cohomology ring of Hom(Z2, SO(3))+ is generated by the image of the canonical generators of H1(SO(3)2;F2) through the inclusion Hom(Z2, SO(3))+ → SO(3)× SO(3) and it is isomorphic to F2[u, v]/〈u 4, v4, u2v + uv2〉 Proof. Consider the spectral sequence of the fibration Hom(Z2, SU(2))→ Hom(Z2, SO(3))+ → RP ∞ × RP∞ So, from the diagram SU(2) ∨ SU(2) // Hom(Z2, SU(2)) //  SU(2)× SU(2)  SO(3) ∨ SO(3) // Hom(Z2, SO(3))+ //  SO(3)× SO(3)  RP∞ × RP∞ = // RP∞ × RP∞ it can be seen that the action of the fundamental group of RP∞ × RP∞ on the co- homology of the fiber is trivial. It can also be seen that d4(a) = u 4 and d4(b) = v 4. So we have to compute d3(x). As the symmetric group on two letters acts on this fi- bration we see that d3(x) has to be one of the three non-trivial symmetric classes of 24 H3(RP∞ × RP∞;F2). For dimensional reasons we must have d2(y) 6= 0, and thus the non-trivial class in H4(Hom(Z2, SO(3))+;F2) comes from H 4(RP∞ × RP∞;F2). As H1(Hom(Z2, SU(2));Z2) is trivial we see that d4(Sq 1(x)) = 0. Thus, from the relation d4(Sq 1(x)) = Sq1(d3(x)) we get d3(x) = u 2v + uv2, and this completes the proof.  With this latter result and by computing the Bockstein cohomoloy groups (see [14], page 305) we obtain the integral cohomology of Hom(Z2, SO(3))+ Corollary 2.16 The integral cohomology of Hom(Z2, SO(3))+ is given by Hq(Hom(Z2, SO(3))+;Z) =   Z q = 0 Z2 ⊕ Z2 q = 1 Z⊕ Z2 ⊕ Z2 q = 2 Z⊕ Z⊕ Z2 q = 3 Z2 q = 4 0 q ≥ 5 We now discuss the case n = 3. First recall from [1] that there is a sequence of cofibrations that split after a single suspension S3(2, SU(2))→ S3(1, SU(2))→ Hom(Z 3, SU(2)) where Sn(j,G) is the subspace ofHom(Z n, G) whose elements have at least j coordinates equal to 1G. Thus S3(2, SU(2)) = SU(2) ∨ SU(2) ∨ SU(2) and therefore after a single suspension Hom(Z3, SU(2)) is homotopy equivalent to ∨ 3 SU(2) ∨ S3(1, SU(2))/S3(2, SU(2)) ∨ Hom(Z3, SU(2))/S3(1, SU(2)) The second summand in this decomposition is identified by the homeomorphism S3(1, SU(2))/S3(2, SU(2))→ ∨ 3 Hom(Z2, SU(2))/S2(1, SU(2)) = ∨ 3 (S6 − SO(3)) given by the map (q1, q2, q3) 7→ ((q1, q2), (q1, q3), (q2, q3)); whereas the third summand is homotopy equivalent to SU(2) ∧ (S6 − SO(3)) (see [1]). Let a, b, c be the classes 25 in H3(Hom(Z3, SU(2));F2) corresponding to the first summand, and x1, x2, x3 be the classes in H2(Hom(Z3, SU(2));F2) corresponding to the seccond summand. Proposition 2.17 The mod-2 cohomology ring of Hom(Z3, SO(3))+ is generated by the image of the canonical generators of H1(SO(3)3;F2) through the inclusion map Hom(Z3, SO(3))+ → SO(3)× SO(3)× SO(3) and by two classes e4 ∈ H 4 and e5 ∈ H 5, and it is isomorphic to F2[u, v, w, e4, e5] modulo 〈u4, v4, w4, u2v + uv2, u2w + uw2, v2w + vw2, e4u, e4v, e4w, e 2 4, e 2 5, e5u, e5v, e5w〉 Proof. In the spectral sequence of the fibration Hom(Z3, SU(2))→ Hom(Z3, SO(3))+ → RP ∞ × RP∞ × RP∞ we see that d4(a) = u 4, d4(b) = v 4 and d4(c) = w 4. Thus we have to compute d3(xi). The action of the symmetric group on three letters imposes several conditions on the possible values of d3(xi), namely we have to find three homogeneous polynomials f, g, h ∈ F2[u, v, w] of degree three such that σ(f) = h, σ(g) = f , σ(h) = g, τ(f) = g and τ(h) = h, where σ is the 3-cycle (u v w) and τ is the transposition (v w). By Propostion 2.15 we can assume that d3(x1) = u 2v+uv2, and therefore d3(x2) = u 2w+uw2 and d2(x3) = v2w+ vw2. Note that there are fifteen monomials of degree four on u, v, w, and with the help of a, b, c we get rid of three of them, namely u4, v4, w4. Now, the image under d3 of the classes generated by the product of the xi ′s with u2v + uv2, u2w + uw2, v2w + vw2 generate a nine dimensional space. So we are left with a three dimensional space at the coordinate (4, 0) of our spectral sequence. That means that there is a class e4 ∈ H4(Hom(Zn, SO(3))+;F2) that does not come from RP ∞ × RP∞ × RP∞. Note that 26 we only need to compute e4u, e4v and e4w. As u is a 2-torsion class we see that so is any class that represents the cup product of u with e4 in integral cohomology. But the five dimensional class comes from a torsion-free class, therefore e4u = 0, and similarly e4v = e4w = 0. This completes the proof.  In this case the integral cohomology of Hom(Z3, SO(3))+ has higher 2-torsion. This can be seen by computing the fourth and fifth Bockstein cohomology groups and by recalling the fact that the class e5 comes from a torsion-free class. These two imply that there is an integral class in dimension 4 of torsion 2k for some k > 1. We close this section by deriving some results on the mod-2 cohomology of the space Hom(Z2, U(2)). Note that the covering SU(2)× R→ U(2) shows that Hom(Z2, U(2)) is covered by Hom(Z2, SU(2)). Now, consider the fibration Hom(Z2, SU(2))→ Hom(Z2, U(2))→ S1 × S1 defined by the covering Hom(Z2, SU(2)) → Hom(Z2, U(2)). If we look at the 3- dimensional classes of the mod-2 cohomology of Hom(Z2, SU(2)), then we can see that the action of Z×Z = pi1(S 1×S1) on these classes is trivial since they come from different summands in H3(Hom(Z2, SU(2));Z2), namely, two come from S 3 ∨ S3 and one from S6 − SO(3). Thus for dimensional reasons we see that the mod-2 spectral sequence of the fibration Hom(Z2, SU(2))→ Hom(Z2, U(2))→ S1 × S1 collapses at the second term. We obtain the following proposition 27 Proposition 2.18 The mod-2 cohomology of Hom(Z2, U(2)) is given by Hq(Hom(Z 2, U(2));Z2) =   Z2 q = 0 Z22 q = 1 Z22 q = 2 Z52 q = 3 Z82 q = 4 Z52 q = 5 Z2 q = 6 0 q ≥ 7 2.5 Rational homology The rational homology of the spaces Hom(Zn, G) can be obtained following a couple of lemmas due to T. Baird [5] and the Vietoris-Begle Theorem. Suppose that a compact connected Lie group G acts on a compact space X, and suppose that every point in X is fixed by a maximal torus in G. Let T be a fixed maximal torus in G. So the map φ : G × XT → X given by (g, x) 7→ gx is onto. The Lie group G acts on G × XT by g·(h, x) = (gh, x) and φ is G equivariant. The normalizer of T in G, N(T ) = NG(T ), acts on G×XT from the right by (g, x) · n = (gn, n−1x), leaving φ invariant and commuting with the G action. Lemma 2.19 Let x ∈ XT . Then g · x ∈ XT if and only if, g ∈ N(T )Gox, where G o x is the identity component of the centralizer of x. Proof. If g · x ∈ XT , then g−1tg · x = x for all t ∈ T . Thus T g ⊆ G0x. Since T g is a maximal torus in G so is T g in G0x. Thus there is some h ∈ G o x so that T gh = T , and therefore g ∈ N(T )Gox. The converse is immediate.  Lemma 2.20 For every x ∈ X, φ−1(x)/N(T ) is homeomorphic to Gox/NG0x(T ). 28 Proof. We may assume by equivariance that x ∈ XT . Thus φ−1(x) = {(g, y) ∈ G×XT | gy = x} and by the last lemma this is homeomorphic to N(T )Gox. Thus φ −1(x)/N(T ) is homeo- morphic to Gox/NG0x(T ) as required.  Before going on let us recall that if G is a Lie group with Weyl group W , then the quotient G/NG(T ) is Q-acyclic. This follows from the fact that χ(G/NG(T )) = 1 and that H∗(G/T ) is generated by H2(G/T ), thus the |W |-covering G/T → G/NG(T ) shows that H∗(G/NG(T );Q) = H ∗(G/T )W = Q. The following result will be the final ingredient to compute the rational cohomology of Hom(Zn, G). Theorem 2.21 (Vietoris-Begle) Let f : A → B be a map between compact Haus- dorff spaces and let K be a field. If f is onto and the fibers are K-acyclic, then the homomorphism induced by f in homology with coefficients in K is an isomorphism. Theorem 2.22 If G is a Lie group with Weyl group W , then the rational cohomology of X is isomorphic (as rings) to H∗(G/T ×XT ;Q)W . Proof. Notice that we have a commutative diagram G×XT φ &&NN NN NN NN NN NN N  (G×XT )/T //  X (G×XT )/N(T ) 88qqqqqqqqqqqq where the bottom vertical map is a |W |-covering. Note that the fibers of the map (G×XT )/N(T )→ X are Q-acyclic, therefore the result follows from the Vietoris-Begle Theorem.  Notice that in the proof it did not matter if we had taken only a subspace of XT , as long as the map φ was onto. Moreover, this last result also holds for fields of characteristic relatively prime to the order of W . 29 We apply this last result to the case in which X is Hom(Zn, G) and the second factor of the source is T n, where T is the maximal torus of G. Recall that the component in Hom(Zn, G) of the trivial representation is denoted by Hom(Zn, G)+. In [18] it is shown that Lemma 2.23 If T is the maximal torus of G, then the map φ : G/T×T n → Hom(Zn, G)+ is onto. Therefore H∗(Hom(Zn, G)+;Q) ∼= H ∗(G/T × T n;Q)W Example 2.24 In the case of Hom(Zn, G) when G is U(n) or SU(n) we see that if A1, . . . , An ∈ G commute, then there is a C ∈ G so that A C j ∈ T for all j. If we let G act on Hom(Zn, G) by conjugation, then T n ⊂ Hom(Zn, G)T . So the map φ : G×T n → Hom(Zn, G) is onto. Corollary 2.25 The Euler characteristic of Hom(Zn, G)+ is zero. Proof. We know that H∗(Hom(Zn, G)+;Q) = H ∗(G × T n/N(T );Q) and that the Euler characteristic of G × T n/N(T ) is equal to the product of |W | with the Euler characteristic of G/T × T n. But the Euler characteristic of T n is zero. Hence the product is zero as well.  Theorem 2.26 If p : G̃ → G is a finite covering of the Lie group G, then the induced covering Hom(Zn, G̃)+ → Hom(Z n, G)+ induces an isomorphism on rational cohomol- ogy. 30 Proof. Let T̃ = p−1(T ). We have the following commutative diagram G̃/T̃ × T̃ n q uujjjj jjj jjj jjj jj ))SSS SSS SSS SSS SS  (G̃× T̃ n)/N(T̃ ) //  Hom(Zn, G̃)  G/T × T n q ttjjjj jjj jjj jjj jjj ))TTT TTT TTT TTT TTT (G× T n)/N(T ) // Hom(Zn, G) in which the horizontal maps are isomorphisms on rational cohomology and the maps q are covering maps. Note thatWG̃ is isomorphic toWG and that G̃/T̃ is homeomorphic to G/T through p. Moreover, the action of Ker(p) on T̃ is homologically trivial, hence the middle vertical map in the diagram induces a rational isomorphism on the W -invariant subrings. The result follows.  Corollary 2.27 The rational cohomology of Hom(Zn, SU(2)) is given by the isomor- phism induced by the map Hom(Zn, SU(2))→ Hom(Zn, SO(3))+ induced by the double covering SU(2)→ SO(3). This last result was obtained independently in [5]. 2.6 Noncommuting elements In this section we will focus on the the complement of Hom(Zn, G) in Gn, that is, the space of noncommuting elements in G, and we will study some geometric properties for the case G = SO(3) and U(2). Definition 2.28 The space of (ordered) noncommuting n-tuples of elements from G is denoted by Xn(G) = G n − Hom(Zn, G) . In the case in which Hom(Zn, G) is not connected we will denote by Yn(G) the complement of the component of the trivial rep- resentation Hom(Zn, G)+ in G n. 31 2.6.1 The case of SO(3) We begin by studying the complement of the spaceHom(Zn, SO(3))+, that is, Yn(SO(3)). As Hom(Zn, SO(3))+ consists of rotations about a common axis we see that Yn(SO(3)) consists of n-tuples of rotations about at least two different axes. Recall that a rotation is determined by a line in RP 2 and an angle in S1 (+ some identifications), and that the identity is a rotation about any line. Thus Y2(SO(3)) is homotopy equivalent to F (2,RP 2) the space of configurations of two points in RP 2, and this is homotopy equiv- alent to V2(R 3)/Z2⊕Z2 which is S 3/Q8. Let Yij be the space of n-tuples of rotations such that the axis of rotation of the ith-entry is different from the axis of rotation of the jth- entry. Thus Yij is homemorphic to Y2(SO(3))×SO(3) n−2. Recall thatHom(Zn, SO(3))+ is covered by Hom(Zn, SU(2)), so we have 2n-covers Xn(SU(2))→ Yn(SO(3)) Xij → Yij where Xij = {(A1, . . . , An) ∈ SU(2) n | [Ai, Aj ] 6= I}. We will analyze the case n = 3, Y3(SO(3)) = Y12 ∪ Y13 ∪ Y23. Our goal is to show that pi1(Y3(SO(3))) is isomorphic to Z2 ×Q8. More generally, we want to show the following Theorem 2.29 One has pi1(Y3(SO(3))) ∼= Z2×Q8 and pi1(Yn(SO(3))) ∼= Z n 2 for n ≥ 4. Furthermore H3n−2(Hom(Zn, SU(2));Z) ∼= H1(Xn(SU(2));Z) =   Z2 n = 2, 30 n ≥ 4 Remark 2.30 The nontrivial class in H7(Hom(Z3, SU(2));Z) = Z2 was computed in [1]; this class is rather mysterious and may be regarded as a pathology of the space Hom(Z3, SU(2)) which is locally euclidean of dimension five except at eight points. This result shows that this type of class appears only when n = 3. 32 Let Y123 = Y12 ∩ Y13 ∩ Y23. Note that Y123 is homotopy equivalent to F (3,RP 2), the configuration space of three points in RP 2; and in general if we let Y12···n be the inter- section of the ( n 2 ) pieces, then it is homotopy equivalent to F (n,RP 2), the configuration space of n points in RP 2. We have the following commutative diagram RP 2 − {∗, ∗′}  // RP 2 − {∗}  // SO(3)  Y123 p12  // Y12 ∩ Y13 p12  // Y12 p12  Y2(SO(3)) = // Y2(SO(3)) = // Y2(SO(3)) where p12 projects the first and second entries. Thus the top row in this diagram becomes S1 ∨ S1 → S1 → SO(3), and it is easy to see that the first map sends each copy of S1 to once around S1 while the second map is the generator of pi1(SO(3)). The map p12 admits a section given by s(A,B) = (A,B,AB) where AB is a rotation about the cross product between the rotation axes of A and B. We claim that pi1(Y12 ∩ Y13) is Z×Q8. To see this we look at the following diagram, which shows that the action of Q8 on Z is trivial. S1 //  S1 × S1 proj //  S1  S1  // Y12 ∩ Y13 p12 // p1  Y2(SO(3)) p1  {∗} // SO(3)− {I} = // SO(3)− {I} To compute pi1(Y123) one has to study the fibration S 1 ∨ S1 → F (3,RP 2)→ S3/Q8 and the following diagram S1 ∨ S1 //  F (2,RP 2 − {∗})  // S1  S1 ∨ S1 //  Y123 // p1  Y2(SO(3)) p1  {∗} // SO(3)− {I} = // SO(3)− {I} 33 A presentation of pi1(Y123) has been given in [13], however we will not need it for our purposes and we only will write this group as (Z ∗ Z)×λ Q8. To compute pi1(Y3(SO(3))) we will use the following diagrams Y12 ∩ Y13 //  Y13  Y12 // Y12 ∪ Y13 Y123 //  Y13 ∩ Y23  Y12 ∩ Y23 // (Y12 ∪ Y13) ∩ Y23 (Y12 ∪ Y13) ∩ Y23 //  Y23  Y12 ∪ Y13 // Y3(SO(3)) By applying pi1 to these latter diagrams and using the diagrams of their corresponding fibrations, we see that they become (respectively) Z×Q8 //  Z2 ×Q8  Z2 ×Q8 // Z2 ×Q8 (Z ∗ Z)×λ Q8 //  Z×Q8  Z×Q8 // Z×Q8 Z×Q8 //  Z2 ×Q8  Z2 ×Q8 // Z2 ×Q8 Hence pi1(Y3(SO(3))) ∼= Z2 ×Q8 as wanted. 34 For the case n ≥ 4, we can see that diagrams of the form Yij ← Yij ∩ Yik → Yik become Zn−22 ×Q8 ←− Z× Z n−3 2 ×Q8 −→ Z n−2 2 ×Q8 where the maps are of the form r × 1 Z n−3 2 × 1Q8, with r : Z→ Z2 the natural reduction map. The pushout of this diagram is isomorphic to Zn−22 ×Q8. As n ≥ 4 there may be diagrams of the form Yij ← Yij ∩ Ykl → Ykl which become Z22 ×Q8 × Z n−4 2 t1←− Q8 ×Q8 × Z n−4 2 t2−→ Q8 × Z 2 2 × Z n−4 2 where the maps are t1 = a × 1Q8 × 1Zn−42 and t2 = ×1Q8 × ×1Z n−4 2 , with a : Q8 → Z 2 2 the abelianization map. The push out of this diagram is isomorphic to Zn2 . As the fundamental group of Yn(SO(3)) can be obtained by computing the direct limit of the pi1(Yij) it follows that pi1(Yn(SO(3)) = Z n 2 as wanted. 2.6.2 The case of U(2) The aim of this section is to prove the following Theorem 2.31 The direct product of the fundamental group of the Klein bottle with Z2 acts properly and discontinuously on S3 × R2 in a non-trivial way. Remark 2.32 Of course the the group mentioned in this result acts on S3 × R2 since R2 is the universal covering of the Klein bottle. What this result claims is that there is an action of this group which is not a product action. Recall the map SU(2)×R p → U(2) given by (A, t) 7→ (epiitI2)A is the universal cover of U(2). Proposition 2.33 The universal cover of X2(U(2)) has the homotopy type of S 3. 35 Proof. As the elements epiitI2 are central in U(2) we see that SU(2)×R×SU(2)×R p×p → U(2)2 induces the cover X2(SU(2))× R 2 → X2(U(2)) with fiber Z2. Since X2(SU(2)) ' RP 3 it follows that the universal cover ofX2(SU(2))× R2 has the homotopy type of S3, and so the result follows.  Proposition 2.34 X2(U(2)) has the homotopy type of ∂ −1(−I) where ∂ : U(2)2 → U(2) is the commutator map. Proof. It is easy to see that ∂−1S3 (−I) × R 2 is a cover of ∂−1(−I) where ∂S3 is the restriction of ∂ to S3. Recall the inclusion ∂−1S3 (−I) → X2(SU(2)) is a homotopy equivalence, so the result follows from the following commutative diagram of covering spaces ∂−1S3 (−I)× R 2 //  X2(SU(2))× R 2  ∂−1(−I) // X2(U(2)) 2 We can also prove this last proposition as follows. Note that the commutator of two elements in U(2) lies in SU(2) and thus the map ∂ : U(2)2 → SU(2) is onto. In [15] it is shown that the critical points of ∂ are the reducible representations and thus ∂−1(I) is contained in this set. It follows that ∂ yields a submersion X2(U(2))→ S 3−{I} and therefore it is a fibration with fiber ∂−1(−I). So we have proved that X2(U(2)) has the homotopy of a 5-dimensional closed manifold and that Γ = pi1(X2(U(2))) acts properly and discontinuosly on S3 × R2. Note also that the group Z acting on the total space of the cover SU(2)× R → U(2) is generated by (−I2, 1), and thus the action of Z × Z on X2(SU(2)) × R 2 is not concentrated only on the R2 factor. We will now focus on computing Γ. In order to compute Γ we start by considering the space Hom(Z2, U(2)). We know that this space is connected and that there are only two kinds of elements in U(2), the 36 regular elements and the central elements. The regular elements have (a conjugate of) T 2 as centralizer. So we get a fibration: U(2)− T  X2(U(2)) p1  U(2)− Z(U(2)) Now we can let Z(U(2))×Z(U(2)) act on X2(U(2) by z, z ′) · (a, b) = (za, z′b), where Z(U(2)) is the center of U(2) and it is isomorphic to S1. This action gives rise to two different actions, namely: one of Z(U(2)) on the fiber U(2)− T which is the restiction of the action of Z(U(2))×Z(U(2)) to 1×Z(U(2)); and one more action of Z(U(2)) on the base U(2)− Z(U(2)) given by left multiplication. These actions yield the following commutative diagram S1 //  U(2)− T  // U(2)− T/S1  S1 × S1 // p1  X2(U(2)) //  X2(U(2))/S 1 × S1  S1 // U(2)− Z(U(2)) // U(2)− Z(U(2))/S1 in which all rows and the first two columns are fibrations. So our task will be to identify each space in this latter diagram. Recall that PU(2) = SO(3) as can be seen in the following diagram Z2 //  S1 //  S1  S3 //  U(2) det //  S1  RP 3 ∼= // PU(2) // {∗} Thus we see that U(2)− T/Z(U(2)) = RP 3 − RP 1. 37 Lemma 2.35 We have the following homotopy equivalences 1. U(2)− Z(U(2))/S1 ' RP 2 2. U(2)− T ' S1 × S1 3. U(2)− T/S1 ' RP 3 − RP 1 ' S1 Proof. For (1) recall that PU(2) = SO(3), while U(2) − Z(U(2))/S1 is precisely PU(2) taking off the class of the identity, so it is PU(2) taking off a point and this is homotopy equivalent to RP 2. To study the topology of U(2)− T we can use the fibrations U(1) = //  U(1)  U(2)− T //  U(2) proj  S3 − S1 res // S3 Recall S3 − S1 ' S1, so we get that U(2)− T is a K(Z2, 1). We claim that RP 3 − RP 1 ' S1. The 2-cover S3 → RP 3 induces a 2-cover S3 − S1 → RP 3 − RP 1 which implies that piq(S 1) = piq(RP 3 − RP 1) for q ≥ 2. Recall U(2)−T ' T , so pi1(U(2)−T ) is abelian and thus pi1(RP 3−RP 1) is either Z or Z⊕Z2. As U(2)− T ' T its cohomology is trivial in degrees ≥ 3. If H1(RP 3−RP 1) = Z⊕Z2, then H2(RP 3−RP 1;Z2) 6= 0, but this would generate a non-trivial cohomology class of degree 3 in H∗(U(2)− T ;Z2), which is not possible. Hence RP 3 − RP 1 is a K(Z, 1) as wanted. Note that the fibration S1 → U(2) − Z(U(2)) → RP 2 corresponds to a class in H2(RP 2) = [RP 2,CP∞] = Z2, so it is the pullback of the Hopf fibration over S 2, and so its transgression is known. Explicitly, the nontrivial map in [RP 2,CP∞] is homotopic to RP 2 ↪→ RP 3 → CP 1 = S2, where the second map is the quotient of the identity in S3 after identifying by the action of Z2 and S 1.  38 Lemma 2.36 X2(U(2))/S 1 × S1 ' S3/Q8 Proof. If Z(G) is the center of G, then Z(G)n acts freely on Hom(Zn, G) and on Gn − Hom(Zn, G) by left multiplication. It is easy to check that Z2 × Z2 acts on X2(SU(2)) with orbit space S 3/Q8. When G = U(2), then we can look at the isomorphism S3 × S1/Z2 → U(2) given by (A, z) 7→ (zI2)A, where Z2 is generated by (−I,−1). Hence the center of U(2) under this isomophism corresponds to {(±I,±z) | z ∈ S1}. So X2(U(2)) is homeo- morphic to X2(SU(2)) × S 1 × S1/Z2 × Z2 with (−1, 1) · (A, z,B,w) = (−A,−z,B,w) and (1,−1) · (A, z,B,w) = (A, z,−B,−w). So we see that the action of Z(U(2)) × Z(U(2)) = S1 × S1 on X2(U(2)) corresponds on the action of S 1/Z2 × S 1/Z2 on X2(SU(2)) × S 1 × S1/Z2 × Z2 given by ([I, λ], [I, 1]) · [A, z,B,w] = [A, λz,B,w] and ([I, 1], [I, λ]) · [A, z,B,w] = [A, z,B, λw]. Analogously, ∂−1U(2)(−I) is homeomorphic to ∂−1S3 (−I)×S 1×S1/Z2×Z2, and the action of Z(U(2))×Z(U(2)) on X2(U(2)) restricts to ∂−1U(2)(−I). Note that ([I, z −1], [I, w−1]) · [A, z,B,w] = [A, 1, B, 1], so ∂−1U(2)(−I)/S 1 × S1 is homeomorphic to S3/Q8, and hence X2(U(2))/S 1 × S1 ' S3/Q8.  Therefore our former diagram turns into the following diagram S1 //  T //  S1  T // p1  X2(U(2)) //  S3/Q8  S1 // U(2)− Z(U(2)) // RP 2 in which we may assume that all rows and columns are fibrations by the following lemma. 39 Lemma 2.37 If we have a commutative diagram F1 //  E1 //  B1  F2 //  E2 //  B2  F3 // E3 // B3 in which all arrows and columns are fibrations except for the right column, then the homotopy fiber of B2 → B3 has the homotopy type of B1. Proof. The construction used in Lemma 2.1 of [9] allows us to replace our diagram by a diagram in which all columns and rows are fibrations. It is easy to see that the top row of the original diagram maps into the top row of this auxiliary diagram, and that this mapping defines a homotopy equivalence between the total spaces and the fibers. The result follows.  At this point we only need to compute pi1(U(2)− Z(U(2))). By duality we have the isomorphism Hq(U(2), U(2)− Z(U(2))) = H 4−q(S1) thus from the long exact sequence of the pair (U(2), U(2) − Z(U(2))) we get that H1(U(2) − Z(U(2))) = Z. As the center of U(2) acts freely on U(2) − Z(U(2)) by left multiplication, we get a fibration S1 = //  S1  U(2)− Z(U(2)) //  U(2)  U(2)− Z(U(2))/S1 // PU(2) Since pi2(PU(2)) = 0 it follows that there is a short exact sequence as follows 1→ Z→ pi1(U(2)− Z(U(2)))→ Z2 → 1 40 It is easy to check that pi1(U(2)−Z(U(2))) = Z by using the fact that its abelianization is Z and that the map H1(S 1)→ H1(U(2)−Z(U(2))) induced by inclusion is nontrivial. The following observations describe the group Γ and allow us to describe it. Remark 2.38 From the diagram that we obtained we can see that 1. The group Γ fits in the following diagram of short exact sequences 1  1  1  1 // Z //  E //  Z/4 //  1 1 // Z2 //  Γ //  Q8 //  1 1 // Z ˆ 2 //  Z //  Z2 //  1 1 1 1 where the group E arises as the quotient the map pi2(U(2)− Z(U(2)))→ pi1(U(2)− T ) and thus it is isomporphic to either: Z⊕ Z/4, Z or Z2 ⊕ Z. 2. The middle column splits and so Γ is isomorphic to E ×λ Z 3. As RP 3 × R2 covers X2(U(2)) we see that Γ has an element of order two. So E cannot be Z otherwise the splitting of the middle column would imply that Γ is a torsion free group. Notice that every torsion element in Γ comes from E, and that E has only one involution (Milnor’s condition). 4. The Hochschild-Serre spectral sequence of 1→ E → Γ→ Z→ 1 collapses 5. The cohomology of Γ and the Farrell cohomology of Γ are periodic (see [10] and [3]) 41 As Γ is a semidirect product of E by Z we can investigate the possible candidates for Γ by studying the maps from Z to the group of automorphisms of E. After doing some routine computations we find that E is Z2 ⊕ Z and that Γ is isomorphic to the group < a, b, c | ab = ba, b2 = 1, c−1ac = a−1, bc = cb > and this group is the direct product of Z2 with the fundamental group of the Klein bottle (see Theorem 2.31). 42 Chapter 3 Group Cohomology 3.1 Filtrations of BG with spaces of homomorphisms In this chapter we will use the spaces of homomorphisms to construct a family of sim- plicial spaces. This family will yield a filtration of the bar construction of the classifying space of a group, and it will be parametrized by the lower central series of a free group. Before defining these spaces let us recall some basic definitions and constructions that can be found for instance in [6]. Let ∆n = {(t0, . . . , tn)|ti ≥ 1, ∑ ti = 1}, the topological n-simplex, and let ∆ be the category of finite ordered sets and non-decreasing maps (an object of ∆ will be denoted by [n] = {0, . . . , n}). The morphisms in ∆ can be generated by injective maps di : [n− 1]→ [n] and surjective maps si : [n+1]→ [n] with 0 ≤ i ≤ n given by di(j) =   j if j < ij + 1 if j ≥ i and si(j) =   j if j ≤ ij − 1 if j > i 43 These morphisms satisfy djdi = didj−1 if i < j sjsi = didj+1 if i ≤ j sjdi = disj−1 if i < j sjdi = 1 = sjdj+1 sjdi = di−1dj if i > j + 1 A simplicial object X in a category C is defined as a contravariant functor from ∆ to C, and it can be thought of as a sequence of spaces Xn, n ≥ 0, and maps di : Xn → Xn−1 and sn : Xn → Xn+1 satisfying the dual identities to the ones displayed above. The object X is also denoted by X∗. The set Xn is called the set of n-simplices of X, and the maps di and si are called the face and degeneracy maps of X respectively. The geometric realization of a simplicial space X is defined as |X| = ⊔ n≥0 ∆n ×Xn/ ∼ where (dix, p) ∼ (x, dip) and (s ix, p) ∼ (x, sip). If C is a category we define the nerve of C as a simplicial set N(C) whose n-simplexes are given by N(C)n = HomCat([n],C) where Cat is the category whose objets are small categories and functors as morphisms, and [n] is thought of as the category 0→ 1→ · · · → n. So an n-simplex is just a length n sequence of composable arrows in C. The geometric realization of the nerve of a category C is often called the classifying space of C. It has many convenient properties, for instance, if G is a topological group and G is the category with only one object and morphisms the elements of G, and if EG is the category whose objects are the elements of G and there is only one morphism between any two objects, then we can see that G acts on the nerve of EG with quotient 44 isomorphic to the nerve of G. Moreover, |N(EG)| is contractible as every object is both final and cofinal. Hence |N(G)| is a model for the classifying space of G. This construction is sometimes called the bar-construction of the classifying space of G, and it is the model that we will use in this work. The transport category Tr(F ) of a functor F : D → Sets is the category whose objects consist of pairs (d, x), where d is an object in D and x ∈ F (d). A map (d, x)→ (d′, x′) is a morphism f : d→ d′ in D such that F (f)(x) = x′. The classifying space of Tr(F ) is called the homotopy colimit of F , and it is denoted by hocolimF . For instance, suppose that X is a G-space and denote by the same letter X the corresponding functor G→ Top, then hocolimX is isomorphic to the Borel construction of the action of G on X. We will now use the spaces of homomorphisms to define simplicial subspaces of the bar-construction of a topological group. Let Fn be the free group generated by e0, . . . , en−1. Let Γ q be the qth-stage of the lower central series of Fn, that is, we define Γq inductively by letting Γ1 = Fn and Γ q = [Fn,Γ q−1]. By convention F0 = {1} and Γq = {1} when q =∞. We define En(q,G) = G×Hom(Fn/Γ q, G) Note that this space can be identified with a subspace of Gn+1. We now define maps di : En(q,G)→ En−1(q,G) and si : En(q,G)→ En+1(q,G), for 0 ≤ q ≤ n, by di(g0, . . . , gn) =   (g0, . . . , gi · gi+1, . . . , gn) 0 ≤ i < n(g0, . . . , gn−1) i = n and si(g0, . . . , gn) = (g0, . . . , gi, e, gi+1, . . . , gn). Similarly, let Bn(q,G) = Hom(Fn/Γ q, G) with maps di and si defined in the same way, except that the first coordinate g0 is omitted and the map d0 takes the form d0(g1, . . . , gn) = (g2, . . . , gn). 45 For 0 ≤ i < n let δi : Fn → Fn+1 be a group homomorphism defined by e0 7→ e0 ... ei−1 7→ ei−1 ei 7→ eiei+1 ei+1 7→ ei+2 ... en−1 7→ en and δn : Fn → Fn+1 be the natural homomorphism defined by the inclusion of the first n generators. Similarly, we define for 0 ≤ i ≤ n a homomorphism σi : Fn+1 → Fn by e0 7→ e0 ... ei−1 7→ ei−1 ei 7→ 1 ei+1 7→ ei ... en 7→ en−1 With this definitions it is easy to see that each di is induced by δi, and that each si is induced by σi. Lemma 3.1 The maps di, si defined on the spaces En(q,G) and Bn(q,G) are well- defined and equip these spaces with a simplicial structure. Proof. The maps di and si are well-defined since they are induced by group homo- morphisms between free groups (the δi’s and σi’s), and from the fact that if f : A→ B is a group homomorphism then f(ΓqA) ⊆ ΓqB. The simplicial structure follows from the definition of the homomorphisms inducing the maps di and si.  46 Note that the map Gn+1 → Gn that projects the last n coordinates onto Gn defines a simplicial map p : E∗(q,G)→ B∗(q,G) Moreover, we can let G act from the left on En(q,G) by multiplication on the first coordinate g(g0, g1 . . . , gn) = (gg0, g1, . . . , gn) and this action makes E∗(q,G) into a G-simplicial space; that is, the action of G com- mutes with the face and degeneracy maps. This action is free and its orbit space is homeomorphic to B∗(q,G). The following definition will be used later and will allow us to understand the local structure of the map p. Definition 3.2 Let FjG n be the subspace of Gn that consists of n-tuples with at least j coordinates equal to e. We define Sn(j, q,G) = Hom(Fn/Γ q, G) ∩ FjG n Proposition 3.3 The map p : |E∗(q,G)| → |B∗(q,G)| is a principal G-bundle. Proof. Let Fn|E∗(q,G)| be the image of∐ k≤n Ek(q,G)×∆k in |E∗(q,G)|, and similarly we define Fn|B∗(q,G)|. The relations imposed by the face and degeneracy maps show that Fn|E∗(q,G)| − Fn−1|E∗(q,G)| = G× (Hom(Fn/Γq, G)− Sn(1, q, G))× (∆n − ∂∆n) whereas Fn|B∗(q,G)| − Fn−1|B∗(q,G)| = (Hom(Fn/Γq, G)− Sn(1, q, G))× (∆n − ∂∆n) The map p restricts to the projection between these subspaces, and so it is a G-bundle as long as the identity of G is a nondegenerate basepoint.  47 Remark 3.4 Notice that the spaces E∗(q,G) and B∗(q,G) are simplicial subspaces of the bar construction of G. In fact, |E∗(∞, G)| = EG, |B∗(∞, G)| = BG and p is the standard map EG→ BG. Proposition 3.5 Let G be a topological group. 1) The natural surjection Fn/Γ q+1 → Fn/Γ q induces a map of simplicial spaces com- patible with the simplicial map pn; that is, the diagram En(q,G) iq // pn  En(q + 1, G) pn  Bn(q,G) iq // Bn(q + 1, G) commutes. 2) There are natural morphisms of principal G-bundles |E∗(q,G)| iq // p  |E∗(q + 1, G)| pn  // EG  |B∗(q,G)| iq // |B∗(q + 1, G)| // BG 3) The maps iq are cofibrations and yield a natural filtration of EG and BG. 4) Suppose that G is a finitely generated discrete group. The filtration is finite if and only if G is nilpotent. In this situation, if G is a nilpotent group of class c, then |E∗(c+ 1, G)| = EG and |B∗(c+ 1, G)| = BG. Proof. Part (1) and (3) are immediate since the maps iq are induced by group homomorphisms, whereas part (3) follows from (2) and from the fact that at the level of simplicial spaces these maps are injective, hence their realization are cofibrations. If G is nilpotent of class c, then Γc+1(G) = {1} and thus Hom(Fn/Γ c+1, G) = Gn for all n. Hence |E∗(c + 1)| = EG and |B∗(c + 1)| = BG. Conversely, if |B∗(q,G)| = BG for some q, then for all n we have Fn|B∗(q,G)| − Fn−1|B∗(q,G)| = Fn|B∗(∞, G)| − Fn−1|B∗(∞, G)| 48 Suppose that G is generated by m elements and that φ : Fm → G is a homomorphism onto the generators of G. We know that Hom(Fm/Γq, G)− Sm(1, q, G) = Hom(Fm, G)− Sm(1,∞, G) so φ ∈ Hom(Fm, G) − Sm(1,∞, G) and thus Γ q(G) = φ(Γq) = {1}. Therefore G is nilpotent.  Definition 3.6 The geometric realizations of E∗(q,G) and B∗(q,G) will be denoted by E(q,G) and B(q,G) respectively. We record some basic properties of the spaces B∗(q,G) and E∗(q,G). Proposition 3.7 Let q ≥ 2 and let G be a topological group. Then the spaces B∗(q,G) and E∗(q,G) define functors from the category of topological groups to the category of simplicial spaces. Proposition 3.8 Let G be a finite group and p be a prime divisor of the order of G. If G has a subgroup H of nilpotency class < q such that (|G : H|, p) = 1, then the map H∗(BG;Fp)→ H ∗(B(q,G);Fp) is injective. Proof. Since B(q,H) = BH the result follows from the following commutative diagram BH = //  BH  B(q,G) // BG in which all the maps are induced by inclusion.  A typical example of this last result is the case in which G has an abelian Sylow p-subgroup. 49 Proposition 3.9 Let G be any topological group, then the homotopy fiber of the inclu- sion map B(q,G)→ BG is E(q,G). Proof. It suffices to look at the following diagram in which every map is a fibration up to homotopy {∗} //  G  G  F //  E(q,G) //  EG  F ′ // B(q,G) // BG  Proposition 3.10 Let G be a finite group. If G is not nilpotent, then there is a q0 that depends on G so that B(q0, G) = B(q,G) for all q ≥ q0. Proof. We define q0 = max{q ≥ 2 | Γ q(H) = {1} for some H < G} Note that q0 is finite sinceG is not nilpotent. ThusHom(Fn/Γ q0, G) = Hom(Fn/Γ q0+l, G) for all n, l ≥ 0, and so the result follows  Proposition 3.11 Let G be a discrete group. Then 1. FnB(q,G)/Fn−1B(q,G) = Σ n(Hom(Fn/Γ q, G)/Sn(1, q, G)) 2. There is a spectral sequence abutting to H∗(B(q,G)) so that E1r,s = Hr+s(FrB(q,G), Fr−1B(q,G)) and when q =∞ this spectral sequence is exactly the bar spectral sequence of G. 50 3.2 B(2, G) for Lie groups In this section we compute the rational cohomology of B(2, G) when G is a Lie group by using the results from Section 2.5. We start off with the following observation. Lemma 3.12 Let Γ be a finite group acting freely and simplicially on X∗. Then the map |X∗| → |X∗|/Γ is a covering map. Before stating the next result we recall that if G is a connected Lie group, then the space Hom(Zn, G) need not be connected. We denote by Hom(Zn, G)+ the component of the trivial representation. Thus the space Bn(2, G)+ = Hom(Z n, G)+ becomes a simplicial space since the degeneracy and face maps are continuous and induced by group homomorphisms. Theorem 3.13 Let G be a compact, connected Lie group, and let T be a maximal torus of G. The map φ : G/T × T n → Hom(Zn, G)+, given by (g, t1, . . . , tn) → (gt1g −1, . . . , gtng −1), induces an isomorphism H∗(G/T ×BT ;Q)W (G) ∼= H∗(B(2, G)+;Q) which is compatible with the map H∗(BG)→ H∗(BT )W (G). Proof. We identify the space G/T × T n with the product G/T ×Bn(∞, T ), with no simplicial structure on G/T . Thus the map φ becomes a simplicial map. Notice that the Weyl group of G acts freely and simplicially on G/T × T n by (gT, t1, . . . , tn) · w = (gwT,w −1t1w, . . . , w −1tnw) 51 and that the map φ is invariant under this action. So we have the following commutative diagram of simplicial spaces G/T × T n //  Hom(Zn, G)+ (G/T × T n)/W (G) 55kkkkkkkkkkkkkkk Note that the vertical map is equivalent to the map (G × T n)/T → (G × T n)/N(T ). Thus we have an isomorphism H∗(Hom(Zn, G)+;Q) → H ∗((G/T × T n)/W (G);Q) for all n (see Section 2.5). It follows that H∗(B(2, G)+;Q)→ H ∗((G/T ×BT )/N ;Q) is an isomorphism as well, and hence H∗(B(2, G);Q) ∼= H∗(G/T × BT ;Q)W (G) as wanted. The last assertion follows from the commutative diagram G/T ×BT φ // proj  B(2, G)+ inc  BT inc // BG  3.3 B(q,G) and group cohomology In this section we study the spaces B(q,G) and for this we begin with some examples that come from a very particular class of groups and which allow us to understand how we can construct B(q,G) from the data of subgroups of G of nilpotency class < q. Example 3.14 The following is a list of cases in which we explicitly identify B(2, G). 1) Recall that S3 = 〈a, b|a 3 = b2 = 1, bab = a−1〉. The subgroups C(a), C(b), C(ab) and C(a2b) cover Hom(Zn, G) in the sense that it is the union of the nth cartesian product of each of these subgroups. Since the centralizers intersect trivially we see that the simplicial set B∗(2, S3) is the pushout of the corresponding simplicial sets 52 of these subgroups over the simplicial set determined by the trivial group. That is, the diagram B∗(2, {1}) sshhhhh hhhh hhhh hhhh hhhh h wwppp pp pp pp pp  ''NN NN NN NN NN N B∗(2, C2) ++VVVV VVV VVVV VVV VVVV VVVV V B∗(2, C2) ''NN NN NN NN NN N B∗(2, C2)  B∗(2, C3) wwppp pp pp pp pp B(2, S3) is a push-out, where the three spaces with C2 correspond to b, ab and a 2b, while the last one to a. As the geometric realization functor commutes with push-outs (see [11]) it follows that B(2, S3) ' B(C2 ∗ C2 ∗ C2 ∗ C3) 2) The set Hom(Zn, Q8) is covered by the simplicial sets determined by the centralizers of the three distinct elements of order four in Q8, but these intersect at the simplcial set of the center of Q8. So we have the diagram B(2, C2) xxqqq qq qq qq q  &&MM MM MM MM MM B(2, C4) &&MM MM MM MM MM B(2, C4)  B(2, C4) xxqqq qq qq qq q B(2, Q8) 3) In the alternating group A4 there are 8 elements of order 3 and three elements of order 2. These last three elements generate a normal subgroup K isomorphic to C2×C2, and of course this group is self-centralizing. As in A4 there is no subgroup of order 6 we see that the elements of order 3 are self-centralizing. Thus B(2, A4) ' B(K ∗ 4C3) where 4C3 is the free product of C3 with itself four times. 53 4) In the dihedral group D8 = 〈a, b|a 4 = b2 = 1, bab = a−1〉 the centralizers are as follows – Z(D8) = {1, a 2} – C(a) =< a > of order 4 – C(b) = C(a2b) and C(ab) = C(a3b), both isomorphic to C2 × C2 Since the intersection of these subgroups is the center of G we have the following diagram B(2, C2) wwnnn nn nn nn nn n  ''PP PP PP PP PP PP B(2, C2 × C2) ''PP PP PP PP PP PP B(2, C4)  B(2, C2 × C2) wwnnn nn nn nn nn n B(2, D8) 5) In general, in the dihedral group D2n = 〈a, b|a n = b2 = 1, bab = a−1〉 the conjugacy classes are as follows: (ar)D2n = {ar, a−r}, (b)D2n = {a2tb| t ≥ 0}, (ab)D2n = {a2t+1b| t ≥ 0} If n is odd then there is only one conjugacy class of elements of order two. If n is even, then Z(D2n) = 〈a n/2〉. The centralizers are as follows: If n = 2k + 1, then C(ar) = 〈a〉, C(b) = 〈b〉, C(ab) = 〈ab〉 If n = 2k, then C(ar) =   D2n if r = k〈a〉 if r 6= k 54 C(b) = {1, b, akb, ak}, C(ab) = {1, ab, ak+1b, ak} Thus, if n is odd B(2, D2n) ' ∨ n BC2 ∨BCn 6) In the alternating group A5 there are 20 elements of order 3, 24 elements of order 5 and 15 of order 2. The following are key properties that will allow us to make our analysis. – If α is a 5-cycle, then it has 24 conjugates in S5 and so we see that CS5(α) has order 5. Thus CS5(α) =< α >⊆ A5 and hence CA5(α) =< α >. Therefore the elements of order 5 are self-centralizing. – The elements of order 3 are conjugate in A5, so CA5(α) has order 3 for any 3-cycle in A5. Therefore the elements of order 3 are self-centralizing. – All the elements of order 2 in A5 are conjugate in A5, so |CA5(α)| = 4 for any α ∈ A5 of order 2. As these groups are the Sylow 2-subgroups of A5 and there is a copy of C2×C2 in A5 we see that all of them are isomorphic to that group. Furthermore, the number of Sylow 2-subgroups of A5 is 5 and they intersect trivially; for if P,Q ∈ Syl2(A5) and there is a nontrivial α ∈ P ∩Q, then P is a proper subgroup of CA5(α), which is a contradiction. With these observations we obtain that B(2, A5) ' B(5(C2 × C2) ∗ 6C5 ∗ 10C3) where nH is the free product of H with itself n-times. Note that in this case we are obtaining the number of Sylow subgroups of the group. We have been working with a special type of groups. The following lemma will tell us why everything has worked so well in the previous examples. 55 Lemma 3.15 Let G be a nonabelian group. The following are equivalent a) G has abelian centralizers. b) If [g, h] = 1, then C(g) = C(h) whenever g, h /∈ Z(G). c) If [g, h] = 1 = [h, k], then [g, k] = 1 whenever h /∈ Z(G). d) If A,B ≤ G and Z(G) < CG(A) ≤ CG(B) < G, then CG(A) = CG(B). Proof. (a)⇒ (b)⇒ (c) is immediate. (c) ⇒ (d) Let a ∈A, g ∈ CG(A) − Z(G), b ∈ B − Z(G), and h ∈ CG(B). We have [b, g] = 1 = [g, a], so [b, a] = 1. As [h, b] = 1 = [b, a] it follows that [h, a] = 1, hence CG(B) ⊆ CG(A) as wanted. (d)⇒ (a) Let x ∈ G−Z(G) and g, h ∈ CG(x). Thus Z(G) < CG(x, g) ≤ CG(x) < G, thus CG(x, g) = CG(x) and hence [g, h] = 1.  Definition 3.16 A finite group satisfying any of the conditions of Lemma 3.15 is called a transitively commutative group, or simply a TC group. As we saw in Example 3.14 the centralizers of elements in G determine the way B(2, G) is constructed. More precisely, the centralizers of elements in G and their intersections determine the structure of B(2, G). This family of subgroups can be given the structure of a poset as follows. Let G be a finite group, say G = {a1, . . . , an}, and let L be the family of subgroups of G of the form CG({ai1, . . . , air}). Recall the following basic properties: a) CG(x) = CG(< x >). b) CG({x, y}) = CG(x) ∩ CG(y) = CG(< x, y >). 56 With these properties we see that L forms a lattice, which is known as the lattice of centralizers of G. Note that if G is a TC group then Lemma 3.15 shows that this lattice looks like G II II II II II uu uu uu uu uu C(a1) HH HH HH HH H · · · C(an) vv vv vv vv v Z(G) Let a1, . . . , ak ∈ G− Z(G) be a set of representatives of their centralizers so that G = ⋃ 1≤i≤k CG(ai) and no smaller number of centralizers covers G. Note that Lemma 3.15 shows that the groups defining this union do not depend on the choice of representatives. We call this number k the number of centralizers that cover G. Theorem 3.17 Let G be a TC discrete group. Then B(2, G) is determined by the lattice of centralizers of G, that is B(2, G) is the push-out of the realisation of the spaces obtained by applying the functor B∗(2, ) to that lattice. Proof. Note that the lattice of centralizers L defines a lattice of simplicial spaces where each vertex is of the form B∗(2, C(ai)); that is, we have the following diagram B∗(2, Z(G)) ((QQ QQ QQ QQ QQ QQ vvmmm mm mm mm mm m B∗(2, CG(a1)) ((QQ QQ QQ QQ QQ QQ · · · B∗(2, CG(ak)) vvmmm mm mm mm mm m B∗(2, G) in which each map is induced by inclusion, and Lemma 3.15 shows that this diagram is a push-out. The result follows from the fact that the geometric realization commutes with push-outs (see [11]).  57 Corollary 3.18 If G is a TC group then B(2, G) and E(2, G) are K(pi, 1)’s. Proof. By the Van Kampen theorem the fundamental group of B(2, G) is the free product of the centralizers that coverG amalgamated along the center ofG. An inductive argument using the Mayer-Vietoris sequence shows that the universal covering X of B(2, G) satisfies Hi(X) = 0 for i > 1. Thus B(2, G) is a K(pi, 1) and so is E(2, G) by Proposition 3.9.  Remark 3.19 Note that this last result can be rephrased by saying that if G is a TC group then B(2, G) = B ( colim A∈A A ) where A is the poset of abelian subgroups of G. The following result can be proved by using trees (see for instance Corollary A.2 from [8] or [21] p. 56). Lemma 3.20 Let F be a subgroup of a free product of groups amalgamated along a common subgroup to all of the factors. If F intersects trivially with every conjugate of the factors, then F is free. Corollary 3.21 If G is a TC group then the fundamental group of E(2, G) is a free group of rank 1− |G : Z(G)|+ (∑ 1≤i≤k |G : Z(G)| − |G : CG(Ai)| ) Proof. Let C be the free product of the centralizers of G amalgamated along the center of G. Notice the fundamental group of E(2, G) is the kernel of the homomorphism between C to G induced by the natural inclusions, and no element in pi1(E(2, G)) is conjugate to an element in a centralizer or else the image of this element is nontrivial in G. To find the rank we calculate the Euler characteristic of the groups in the extension 1→ pi1(E(2, G))→ C → G→ 1 58 Thus rank(pi1(E(2, G)) = 1− χ(pi1(E(2, G))) = 1− χ(C) χ(G) Since C is the amalgamated product of the centralizers covering G along Z(G), it follows by an inductive argument that χ(C) = χ(Z(G)) + k∑ i=1 χ(CG(ai))− χ(Z(G)) As G is finite it follows that 1− χ(C) χ(G) = |G| ( 1 |Z(G)| + ∑ 1≤i≤k 1 |CG(Ai)| − 1 |Z(G)| ) ) The result follows.  In the context of graphs, if C = ∗Z(G) Ci where Ci = CG(ai), then one can construct a graph X by setting Edges(X) = k−1∐ C/Z(G) V ertices(X) = ∐ 1≤i≤k C/Ci and where the ith copy of C/Z(G) is identified with the vertices C/CiunionsqC/Ci+1 by using the natural maps C/Z(G) → C/Ci. It turns out that X is a tree (see [21] p. 38), thus F = pi1(E(2, G)) acts freely on X since it intersects trivially with any conjugate of the Ci’s. Therefore F = pi1(X/F ) and so it is a free group of rank 1− χ(X/F ) = 1− |V ertices(X/F )| + |Edges(X/F )| = 1− ∑ 1≤i≤k |F \ C/Ci|+ (k − 1)|F \ C/Z(G)| = 1− ∑ 1≤i≤k |G/Ci|+ (k − 1)|G/Z(G)| = 1− |G/Z(G)|+ ∑ 1≤i≤k (|G/Z(G)| − |G/Ci|) 59 Note that X/F is a graph, so if we now look at the augmented chain complex of X/F then we get the following exact sequence of G-modules 0→ H1(E(2, G))→ k−1⊕ Z[G/Z(G)]→ ⊕ 1≤i≤k Z[G/Ci]→ Z→ 0 where the structure of H1(E(2, G)) as a G-module is exactly the one defined by the extension 1→ pi1(E(2, G))→ C → G→ 1. Example 3.22 Let F21 =< x, y|x 7 = x3 = 1, xy = yx2 >, the Frobenius group of order 21, then F21 is a TC-group with trivial center and in this case C = C7 ∗ 7C3, thus pi1(E(2, F21)) ∼= F96. For the dihedral group D2n with n odd we obtain rank(pi1(E(2, D2n))) = n 2 − 1; and if n is even, say n = 2k, we obtain rank(pi1(E(2, D2n))) = k 2 − 1. One natural task is to find a basis for pi1(E(2, G)), and this can be achieved in the following way. Consider the sequence 1→ pi1(E(2, G))→ pi1(B(2, G))→ BG→ 1 and suppose that G is a TC group. Then we can write this sequence as 1→ F → C → G → 1 where F is a free group (of known rank) and C is the amalgamated product of the centralizers covering G along Z(G). To compute a basis for F we can use the following algorithm. 1. Suppose that C = 〈X|R〉. Then let K be the free group on X and let H be the preimage of F under the natural map K → C. Then H is a normal subgroup of K and is a free group of rank r = (n− 1)m+ 1, where n = |X| and m = |G|. By enumerating cosets (or other method) find a transversal for H in K. Denote it by U = {u1, · · · , ur} 1 // H //  K //  G =  // 1 1 // F // C // G // 1 60 2. If u ∈ U and x ∈ X, then let ux be the element of U corresponding to the coset Hux. Then, by Nielsen-Schreier (see [17]), the set Y = {ux · ux−1|u ∈ U, x ∈ X} \ {e} is a set of free generators of H. 3. Note that R, the normal closure of R in K, is contained in H and thus F = H/R. As R is generated by elements of the form rf with r ∈ R and f ∈ K, if we write f = uh with u ∈ U and h ∈ H, then R is the normal closure in H of the set S = {ru |r ∈ R, u ∈ U}. If we write this set in terms of the set Y , say S = S(Y ), then we get a presentation for F , namely 〈Y |S(Y ) = 1〉 Example 3.23 Consider the group S3 = 〈s, t|s 3 = t2 = 1, tst = s2〉, for which C = C2 ∗ C2 ∗ C2 ∗ C3. So take X = {a, b, c, d} and R = {a 2, b2, c2, d3}. The group K is precisely the kernel of the homomorphism given by a 7→ t b 7→ st c 7→ s2t d 7→ s Thus a transversal for H in F is U = {1, a, b, c, d, d2}. Then the set Y is a2 ab(d2)−1 acd−1 adc−1 bad−1 b2 bc(d2)−1 bda−1 ca(d2)−1 cbd−1 c2 cdb−1 dab−1 dbc−1 dca−1 d3 d2ac−1 d2ba−1 d2cb−1 61 When we take into account the relations coming from C this latter list becomes abd acd−1 adc bad−1 bcd bda cad cbd−1 cdb dab dbc dca d−1ac d−1ba d−1cb We easily see that this list reduces to x1 = abd x2 = acd −1 x3 = adc x4 = bad −1 x5 = bcd x6 = bda x7 = cad x8 = cbd −1 Hence K is a free group on {x1, . . . , x8}. By performing some routine computations we see that the action of S3 on K is determined by the matrices s←→   0 0 0 −1 −1 −1 0 0 0 0 0 0 1 0 0 0 0 0 −1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 −1 −1 0 0 −1 0 0 0 0 0 −1 −1 −1 0 0 1 0 0 1 0 0 0 0 0 0 0   62 t←→   0 0 0 1 1 1 0 0 0 0 −1 0 0 0 1 1 0 −1 0 −1 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 −1   and in fact this is a faithful representation of S3 in GL8(Z). If we call this representation ρ, then χρ(1) = 8, χρ(t) = −2 and χρ(s) = −1; thus by using the character table of S3 we see that ρ = 2λ2 ⊕ 3λ3, where λ2 is the representation defined by the sign of a permutation and λ3 is the irreducible representation of S3 of dimension 2. Remark 3.24 The representation G → Aut(H1(E(2, G))) for G a TC group need not be faithful. For instance, if G = Q8 then the rank of F = pi1(E(2, G)) is 3 but GL(3,Z) does not have a copy of Q8 ( [24]). We now will focus more on the algebraic structure of TC groups. The following lemma, for instance, will allow us to obtain an explicit description of B(2, G) for TC- groups with trivial center. Lemma 3.25 Let G be a TC group with trivial center. a) The Sylow subgroups of G are abelian and intersect trivially. b) {CG(x)| x 6∈ Z(G)} is the family of maximal abelian subgroups of G. c) The maximal abelian subgroups of G intersect trivially. d) If H is a maximal abelian subgroup and P ∈ Sylp(H), then P ∈ Sylp(G). 63 Proof. For (a), let P ∈ Sylp(G) and 1 6= x ∈ Z(P ), thus P ⊆ CG(x) and so P is abelian. Let P,Q ∈ Sylp(G) so that there is 1 6= x ∈ P ∩Q. So P,Q ⊆ CG(x) and thus PQ is an abelian p-subgroup of G of order |PQ| = |P ||Q| |P ∩Q| > |P | which is a contradiction. Part (b) and (c) are immediate. For (d), let P ∈ Sylp(H) and Q ∈ Sylp(G) so that P ⊆ Q. Let 1 6= x ∈ P , so H,Q ⊆ CG(x) and thus HQ is an abelian subgroup. It follows that Q ⊆ H since H is a maximal abelian subgroup. The result follows.  Remark 3.26 Note that Part (b) of the previous lemma holds for any TC group. With the notation of Theorem 3.17 we have the following result Corollary 3.27 If G is a TC group with trivial center, then B(2, G) ' ∨ 1≤i≤k   ∏ p||CG(ai)| BP   where P ∈ Sylp(G) and np,i is the number of Sylow p-subgroups of G contained in CG(ai). Proof. Notice that B∗(2, G) is the join of the simplicial spaces B∗(2, CG(ai)), and by Lemma 3.25 CG(ai) ∼= ∏ p||CG(ai)| P np,i with P ∈ Sylp(G). The result follows.  Corollary 3.28 If G is a TC group with trivial center, then H̄∗(B(2, G)) = ⊕ 1≤i≤k H̄∗(CG(ai)) and in particular H1(B(2, G)) = ⊕ p||G| P np where P ∈ Sylp(G) and np is the number of Sylow p-subgroups of G. 64 Proof. The first part follows from Lemma 3.25. Note that np = np,1 + · · ·+ np,k, so H1(B(2, G)) = ⊕ 1≤i≤k H1(CG(ai)) = ⊕ 1≤i≤k CG(ai) = ⊕ p||G| P np  Example 3.29 We had already obtained the equivalence B(2, A5) ' ∨ 5 B(C2 × C2) ∨ ∨ 6 BC5 ∨ ∨ 10 BC3 and so we get H1(B(2, A5)) = ⊕ 5 (Z2 × Z2)⊕ ⊕ 6 Z5 ⊕ ⊕ 10 Z3 Note that pi1(E(2, A5)) = F854, so we have a faithful representation A5 → Aut(F854). Another example of a TC group with trivial center is SL(2, 8) (see [20]). Its order is 504 = 23 · 32 · 7 with an elementary abelian Sylow 2-subgroup and with a cyclic Sylow 3-subgroup. So H1(B(2, SL(2, 8))) = ⊕ 9 (Z2 × Z2 × Z2)⊕ ⊕ 28 Z9 ⊕ ⊕ 36 Z7 Remark 3.30 TC-groups are classified, see [20]. Some examples include 1. Groups with an abelian normal subgroup of prime index, e.g. dihedral and gener- alized quaternion groups. 2. SL(2,F2n), with n ≥ 2. 3. All nonabelian groups of order < 24. Remark 3.31 Unlike the case of the classifying space of a group, in which BG ' BH if and only if G ∼= H, we may have B(2, G) ' B(2, H) with G 6∼= H. For example, the dihedral group of order 4(22n+2n) and the group SL(2, 2n), with n ≥ 2, have isomorphic centralizer lattices (see [20]). 65 Remark 3.32 Recall that the map cg : BG→ BG induced by conjugation by g ∈ G is homotopic to the identity. It turns out that for B(2, G) this need not be the case. The reason is that B∗(2, G) splits into a lattice of nerves of categories so that conjugation may permute this lattice. Of course, there are some exceptional cases, those in which the centralizers are normal subgroups, e.g. Q8. Nevertheless, it is not true that if g ∈ G, then the map cg : G → G given by cg(x) = g −1xg induces a map homotopic to the identity on B(2, G). For instance, conjugation induces a permutation on the module H∗(B(2, A5);F2) ∼= ⊕ 5 H∗(C2 × C2;F2) An example of a group of smallest possible order and which is not a TC group is S4, since S4 is a group with trivial center and has a Sylow 2-subgroup isomorphic to D8. We will now discuss the construction of B(2, S4). Example 3.33 Recall that S4 has 8 elements of order 3, 6 of order 4, 6 with cyclic structure (12), and 3 with cyclic structure (12)(34). The elements of order 4 and 3 are self-centralizing, and since there are 4 Sylow 3-subgroups it follows that B(2, S4) will have four summands of the form BC3. Notice that a k-tuple of commuting elements from S4 determines an abelian subgroup, and so it suffices to study the poset of abelian subgroups. The poset of abelian 2-subgroups of S4 is given as follows C(12) LL LL LL LL LL C(1234) C(14) LL LL LL LL LL K iii iii iii iii iii iii ii VVV VVV VVV VVV VVV VVV VVV C(1342) qq qq qq qq qq C(12) C(1324) qq qq qq qq qq < (13)(24) > UUU UUU UUU UUU UUU UUU UU < (14)(23) > < (12)(34) > hhh hhh hhh hhh hhh hhh hhh {1} where the letter C stands for the centralizer of an element in S4, the centralizer of each transposition is isomorphic to C2 × C2, and K = {id, (13)(24), (14)(23), (12)(34)} ∼= 66 C2 × C2. So the poset of abelian subgroups of S4 is described by the following diagram • @@ @@ @@ @ • • DD DD DD DD • nn nn nn nn nn nn nn n QQ QQ QQ QQ QQ QQ QQ Q • zz zz zz zz • • ~~ ~~ ~~ ~  PP PP PP PP PP PP PP PP PP PP PP PP PP PP PP  LL LL LL LL LL LL LL LL LL LL LL L • CC CC CC CC CC CC CC CC C • • {{ {{ {{ {{ {{ {{ {{ {{ {  rr rr rr rr rr rr rr rr rr rr rr r  nn nn nn nn nn nn nn nn nn nn nn nn nn nn nn {1} where each • represents a 2-subgroup, and each  a 3-subgroup. Therefore B(2, S4) ' (∨ 4 BC3 ) ∨ [(∐ 3 B(C2 × C2) unionsqBC4 ) unionsqBK ] / ∐ 3 BC2 From this latter description of B(2, S4) it is not clear what kind of space it is. We will address this problem and since the approach that we will consider works for higher q’s we will work in this generality by considering the poset of subgroups of higher nilpotency class to construct B(q,G). 3.4 The homotopy of B(q,G) Definition 3.34 Given a finite group G and q ≥ 2 we define Nq(G) as the poset of nilpotent subgroups of G of class < q. Notice that N2(G) is precisely the poset of abelian subgroups of G, and recall that the higher commutator subgroups satisfy Γq(H) =< [x1, . . . , xq]|xj ∈ H > for any group H. The use of the definition of Nq(G) is inspired by the fact that an element in Bn(q,G) = Hom(Fn/Γ q, G) determines a subgroup of G of class ≤ q − 1, for if (g1, . . . , gn) ∈ Bn(q,G) then Γ q(< g1, . . . , gn >) = {1}. Thus we get the following result. Theorem 3.35 Let G be a finite group and q ≥ 2. Then B(q,G) is constructed as the push-out of the realization of the diagram of spaces obtained by applying the functor 67 B∗(q, ) to Nq(G). That is B(q,G) = colim H∈Nq(G) BH Proof. Note that there is an obvious map of simplicial spaces Bn(q,G)→ ⋃ H∈Nq(G) Bn(∞, H) ⊆ Bn(∞, G) But the realization of this union is precisely the space that results from identifying the realization of each space in this union along their intersections (i. e. the colimit). This defines the desired homeomorphism.  Remark 3.36 We can obtain a similar description for E(q,G). If H ≤ G then let CG(H) be the category whose objects are the elements of G and morphisms the elements of H acting on the objects and composing by using the multiplication of G. Then there is an obvious map En(q,G)→ ⋃ H∈Nq(G) N(CG(H))n ⊆ En(∞, G) which defines the homeomorphism E(q,G)→ colim H∈Nq(G) |CG(H)| The isomorphism in Theorem 3.35 can be expressed more efficiently in the following way. Let Mq(G) be the set of maximal nilpotent subgroups of G of class < q. Then B(q,G) = colim H∈Mq(G) BH Notice that in order to construct this colimit it suffices to consider only the identifications imposed by the pairwise intersections of elements inMq(G) since these generate the same equivalence relation as that generated by all the elements in Nq(G). Example 3.37 Note that the posets Nq(G) (similarly Pq(G)) get bigger as q increases. For example, N2(S4) ( N3(S4), whereas N3(S4) = Nc(S4) for all c ≥ 3. As the maximal 68 subgroups of class < 3 of S4 are isomorphic to either D8 or C3, and every pair of copies of D8 intersect at K, the Klein 4-group, it follows that for q ≥ 3 B(q, S4) = ∨ 4 BC3 ∨B(∗ K 3D8) where the last group is the free product of three copies of D8 amalgamated along K. Now let Pq(G) be the category with objects the set {Mα,Mα ∩Mβ|Mα,Mβ ∈Mq(G)} and with morphisms the set of inclusions {Mα ∩Mβ →Mα|Mα,Mβ ∈Mq(G)} Notice that this category is 1-dimensional in the sense that there are no compositions. We may think of this category as a graph of groups, that is an oriented graph with a group at each vertex and a homomorphism between the two vertices of each edge according to its orientation. For example, if G is a TC group then the graph defined by P2(G) looks like an asterisk with all the arrows pointing outwards. From this graph of groups we can construct a space BPq(G) by 1. putting the classifying space of each group at each vertex of the graph 2. filling in a mapping cylinder for each map induced at the level of classifying spaces at each edge. This construction can be applied to any graph of groups X and thus there is a space BX associated to it. The following result describes this space (see [14] Theorem 1B.11). Theorem 3.38 If all the edge homomorphisms of X are injective, then BX is a K(pi, 1). This result shows that BPq(G) is a K(pi, 1). Note that all the maps at the level of classifying spaces are cofibrations. If Pq(G) is a tree, then BPq(G) is homotopy 69 equivalent to B(q,G) (by collapsing the mapping cylinders). On the other hand, as Pq(G) is 1-dimensional it follows that BPq(G) is the homotopy colimit of the functor F : Pq(G)→ Top that sends a group to its classifying space. We have proved Theorem 3.39 Let q ≥ 2 and G be a finite group such that Pq(G) is a tree. Then 1. the space B(q,G) is a K(pi, 1). 2. there is a natural homotopy equivalence B(q,G) ' hocolimF where F : Pq(G)→ Top is given by H 7→ BH. This latter result requires a strong condition on Pq(G), namely being a tree. It turns out that B(q,G) can also be seen as the homotopy colimit over all the subgroups of class < q, as can be proved from the following lemma. But this does not imply that it is a K(pi, 1). Lemma 3.40 Let D be a functor from a partially ordered set I into a category C. If the map colim i<j D(i)→ D(j) is a cofibration, then the map hocolimD → colimD is a homotopy equivalence. Proof. See [6] page 330.  Theorem 3.41 Let G be a finite group, then hocolim H∈Mq(G) BH ∼ → colim H∈Mq(G) BH 70 3.5 The homology of B(q,G) In this section we study the homology of the space B(q,G). Recall that if K∗ is a simplicial set then one can define ZK∗ as the free abelian group on the simplices of K∗. The face maps of K∗ define boundary maps by the equation ∂n = n∑ i=0 (−1)idi : ZKn → ZKn−1 and so ZK∗ becomes a chain complex. The crux of this construction are the isomorphisms H∗(ZK∗) ∼= H∗(Sing(|K∗|)) ∼= H∗(|K∗|) where Sing(|K∗|) is the singular complex of |K∗|. Let’s consider ZB∗(q,G), which can be thought of as a subcomplex of ZB∗(∞, G), in fact we have ZB∗(q,G) = ⋃ H∈Nq(G) ZB∗(q,H) Note that ZB0(q,G) = Z and ZB1(q,G) = Z[G]. So this chain complex looks like · · · → ZB2(q,G) ∂2→ Z[G] ∂1→ Z where Z[G] is the free abelian group on G and ∂1 = 0 and ∂2(x, y) = y − xy + x. We define for each q ≥ 2 a subgroup of Z[G] by Iq(G) = 〈y − xy + x|Γ q(〈x, y〉) = 1, with x, y ∈ G〉 So I2(G) ⊆ I3(G) ⊆ · · · ⊆ I∞(G), where Γ q = {1} when q =∞. Then we see that H1(B(q,G)) = Z[G]/Iq(G) and so we have a sequence of surjective maps H1(B(2, G))→ H1(B(3, G))→ · · · → H1(BG) 71 Remark 3.42 Note that we can think of ZB∗(q,G) as the union of the chain complexes generated by the maximal subgroups of class < q. Therefore we can use a Mayer-Vietoris spectral sequence to compute H∗(B(q,G)) Corollary 3.43 If G is a finite group and q ≥ 2, then Hi(B(q,G);Z) is finite for all i > 0, and their torsion is only at primes dividing the order of G. Proof. Note that in the Mayer-Vietoris spectral sequence all the groups involved are finite and killed by |G|. Thus the groups obtained from the extensions of the groups in the spectral sequence are killed by an appropriate power of the order of G.  Similarly, we can consider ZE∗(q,G) and in this case ZE0(q,G) = Z, so this complex looks like · · · → ZE2(q,G) ∂2→ ZE1(q,G) ∂1→ Z[G] where ∂1(a, x) = ax − a and ∂2(a, x, y) = (ax, y) − (a, xy) + (a, x). As the natural projection E∗(q,G)→ B∗(q,G) is a simplicial map we get a map of chain complexes and hence one of homology groups. Note that this map on H1 takes the form H1(E(q,G))→ H1(BG) (z, x) 7→ x at the chain level. Similarly, we have a sequence of surjective maps H1(E(2, G))→ H1(E(3, G))→ · · · → H1(EG) Lemma 3.44 H1(BG) 6= 0 if and only if the map H1(E(2, G)) → H1(B(2, G)) is not surjective. Proof. Recall that the composition E(q,G) → B(q,G) → BG is (homotopic to) a fibration. Thus, if H1(E(2, G)) → H1(B(2, G)) is surjective, then H1(BG) = 0. 72 Now suppose that H1(BG) = 0. Then the Serre spectral sequence of the fibration E(q,G) → B(q,G) → BG shows that H1(E(2, G))G → H1(B(2, G)) is surjective, and therefore so is the map H1(E(2, G))→ H1(B(2, G)).  Remark 3.45 The last result holds for all q ≥ 2. In fact, it can also be proved by considering the exact sequence H2(B(2, G)) // H2(G) // H1(E(2, G))G // H1(B(2, G)) // H1(BG) // 0 H1(E(2, G)) OO 66mmmmmmmmmmmmm Moreover, the following are equivalent 1. (Feit-Thompson) Every finite group of odd order is solvable, 2. If G has odd order, then H1(E(2, G))→ H1(B(2, G)) is not surjective, since for groups of odd order the condition H1(G) 6= 0 is equivalent to being solvable, as can be easily seen by induction on the order. Remark 3.46 As we said in Chapter 1, TC groups were studied by Suzuki ( [23]) and these were the first family for which the “odd-order” theorem was proved, and in fact these turned out to be a model for Feit and Thompson in their work on that celebrated theorem. 73 Bibliography [1] A. Adem, F. Cohen. Commuting elements and spaces of homomorphisms. Math. Ann. 338 (2007), no. 3, 587–626. [2] A. Adem, D. Cohen, F. Cohen. On representations and K-theory of the braid groups. Math. Annalen, 326(2003), 515–542. [3] A. Adem, J. Smith. Periodic complexes and group actions. Ann. of Math. (2), 154(2001), no. 2, 407–435. [4] S. Akbulut and J. McCarthy. Casson’s Invariant for Oriented Homology Spheres. Mathematical Notes 36. Princeton, 1990. [5] T. Baird. Cohomology of the space of commuting n-tuples in a compact Lie group. Algebraic & Geometric Topology 7(2007), 737–754. [6] A. K. Bousfield, D. M. Kan. Homotopy Limits, Completions and Localiza- tions. Lecture Notes in Mathematics 304. Springer-Verlag, Berlin-New York, 1972. [7] A. Borel, R. Friedman, J. W. Morgan. Almost commuting elements in compact Lie groups. Mem. Amer. Math. Soc. 57 (2002), no. 747. [8] K. Brown. Cohomology of Groups. GTM, Vol 87. Springer-Verlag , New York 1982. [9] F. Cohen, J. Moore, J. Neisendorfer. The double suspension and exponents of the homotopy groups of spheres. Ann. of Math., 110(1979), 549–565. [10] F. X. Connolly, S. Prassidis. Groups which act freely on Rm×Sn−1. Topology, 28 1989, 133-148. 74 [11] P. Gabriel, M. Zisman. Calculus of Fractions and Homotopy Theory. Ergeb- nisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer-Verlag, New York 1967. [12] W. Goldman. Topological components of spaces of representations. Invent. Math., 93 (1988), no. 3, 557–607. [13] J. Guaschi, D. Lima-Gonçalves. The braid groups of the projective plane. Algebr. Geom. Topol. 4 (2004) 757-780. [14] A. Hatcher. Algebraic Topology. Cambridge University Press. Cambridge, 2002. [15] J. Igusa. On a property of commutator in a unitary group. Mem. Coll. Scvi. Kyoto Univ., Ser A 26 (1950), 45–49. [16] L. C. Jeffrey. Flat connections on oriented 2-manifolds. Bull. London Math. Soc., 37 (2005), 1–14. [17] D. L. Johnson. Presentation of Groups. L. M. S. Lecture Notes 22. Cambridge University Press. [18] V. G. Kac, A. V. Smilga. Vacuum structure in supersymmetric Yang-Mills theories with any gauge group. The Many Faces of the Superworld. 185–234, (World Sci. Publishing, River Edge, NJ, 2000). [19] J. Li. The space of surface group representations. Manuscripta Math., 78 (1993), no. 3, 223–243. [20] R. Schmidt. Subgroup Lattices of Groups. de Gruyter Expositions in Mathe- matics, 14. Berlin, 1994. [21] J. P. Serre. Trees. Springer-Verlag. New York, 1980. 75 [22] N. E. Steenrod. Cohomology Operations. Annals of Math. Studies No 50. Princeton, 1962. [23] M. Suzuki. Group Theory II. Grundlehren der Mathematischen Wissenschaften, 247. Springer-Verlag, New York, 1982. [24] K. Tahara. On the finite subgroups of GL(3,Z). Nagoya Math. J., 41 (1971), no. 3, 169-209. [25] E. Torres-Giese and D. Sjerve. Fundamental Groups of Commuting Ele- ments in Lie Groups. to appear in Bull. London Math. Soc.

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