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Homotopy colimits of classifying spaces of finite abelian groups Okay, Cihan 2014

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Homotopy colimits of classifyingspaces of finite abelian groupsbyCihan OkayA thesis submitted in partial fulfilment of the requirements forthe degree ofDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)The University of British Columbia(Vancouver)April 2014c© Cihan Okay, 2014AbstractThe classifying space BG of a topological group G can be filtered by a se-quence of subspaces B(q,G), q ≥ 2, using the descending central series offree groups. If G is finite, describing them as homotopy colimits is convenientwhen applying homotopy theoretic methods. In this thesis we introduce nat-ural subspaces B(q,G)p ⊂ B(q,G) defined for a fixed prime p. We showthat B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)p asp runs over the primes dividing the order of G. Colimits of abelian groupsplay an important role in understanding the homotopy type of these spaces.Extraspecial p–groups are key examples, for which these colimits turn out tobe finite. We prove that for extraspecial p–groups of rank r ≥ 4 the spaceB(2, G) does not have the homotopy type of a K(pi, 1) space thus answeringin a negative way a question which appears in [1]. Furthermore, we give agroup theoretic condition, applicable to symmetric groups and general lineargroups, which implies the space B(2, G) not having the homotopy type of aK(pi, 1) space. For a finite group G, we compute the complex K–theory ofB(2, G) modulo torsion.iiPrefaceThis dissertation is original, independent work by the author, C. Okay. Aversion of this material is accepted for publication in Algebraic & GeometricTopology.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Classifying spaces and homotopy colimits . . . . . . . . . . . 51.1 Filtrations of classifying spaces . . . . . . . . . . . . . . . . . 51.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 p-local version . . . . . . . . . . . . . . . . . . . . . . . 81.2 The case of finite groups . . . . . . . . . . . . . . . . . . . . . 101.2.1 Homotopy colimits . . . . . . . . . . . . . . . . . . . . 101.2.2 A stable decomposition of B(q,G) . . . . . . . . . . . . 131.2.3 Homotopy types of B(q,G) and B(q,G)p . . . . . . . . 152 Finiteness of G(2) and homotopy type of B(2, G) . . . . . . . 192.1 Colimits of solvable groups . . . . . . . . . . . . . . . . . . . . 192.2 Elementary properties of G(2) . . . . . . . . . . . . . . . . . . 222.3 Finiteness of G(2) . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 Extraspecial p–groups . . . . . . . . . . . . . . . . . . 292.4 Homotopy type of B(2, G) . . . . . . . . . . . . . . . . . . . . 31iv3 Complex K–theory . . . . . . . . . . . . . . . . . . . . . . . . . 363.1 Higher limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.1 Higher limits of the representation ring functor . . . . 443.2 K-theory of B(2, G) . . . . . . . . . . . . . . . . . . . . . . . . 453.2.1 K-theory of B(2, G)p . . . . . . . . . . . . . . . . . . . 464 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1 Homotopy type of B(2, G) . . . . . . . . . . . . . . . . . . . . 494.2 K–theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58vAcknowledgementsI wish to thank Alejandro Adem, for his supervision and patience; Jeff Smithfor pointing out Proposition 3.1.5 and useful conversations; Fred Cohen andJose Go´mez, for commenting on an early version of this manuscript. I wouldlike to thank Ian Allison for his help with any sort of computer issues. I wantto thank my family for their support.viIntroductionThe classifying space BG is an important object in algebraic topology withapplications to bundle theory and cohomology of groups. When G is discrete,it is homotopy equivalent to an Eilenberg–Maclane space of type K(G, 1).A key property of BG is that it classifies principal G–bundles; there is abijection between the set of isomorphism classes of principal G–bundles overa CW–complex X and the set of homotopy classes of maps X → BG. Inthis thesis we study certain subspaces of the classifying space BG.In [1] a natural filtration of the classifying space BG of a topologicalgroup G is introduced:B(2, G) ⊂ · · · ⊂ B(q,G) ⊂ · · · ⊂ B(∞, G) = BG.For a fixed q ≥ 2, let Γq(Fn) denote the q-th stage of the descending cen-tral series of the free group on n generators. Then B(q,G) is the geometricrealization of the simplicial space whose n-simplices are the spaces of homo-morphisms Hom(Fn/Γq(Fn), G). In this thesis we study homotopy-theoreticproperties of these spaces.In Chapter 1 we introduce the spaces B(q,G) and describe them as homo-topy colimits. Furthermore, we define a subspace B(q,G)p ⊂ B(q,G) whichcaptures information specific to a prime p. Let G be a finite group and p aprime dividing its order. Consider the free pro-p group Pn, the pro-p com-pletion of the free group Fn. As noted in [1], the geometric realization of thesimplicial set n 7→ Hom(Pn/Γq(Pn), G) gives a natural subspace of B(q,G),1and is denoted by B(q,G)p. We prove that there is a stable homotopy equiv-alence:Theorem. Suppose that G is a finite group. There is a natural weak equiv-alence∨p||G|ΣB(q,G)p → ΣB(q,G) for all q ≥ 2induced by the inclusions B(q,G)p → B(q,G).Let N (q,G) denote the collection of subgroups of nilpotency class lessthan q and G(q) = colimN (q,G)A. The key observation in [1] is the followingfibrationhocolimN (q,G)G(q)/A→ B(q,G)→ BG(q),which can be constructed using the (homotopy) colimit description ofB(q,G).This raises the following question of whether they are actually homotopyequivalent appears in [1].Question. [1, page 15] If G is a finite group, are the spaces B(q,G) Eilenberg-Mac Lane spaces of type K(pi, 1)?One of the objectives of this thesis is to show the existence of a certainclass of groups for which B(2, G) does not have the homotopy type of aK(pi, 1) space. In Chapter 2 we address this question, and study the colimitsG(q) more closely. The following theorem is used to show that extraspecialp-groups are examples of such groups. Let Ĝ denote the kernel of the mapG×G→ G/[G,G] given by (x, y) 7→ xy[G,G].Theorem. Let Srp denote an extraspecial p-group of rank 2r ≥ 4 thenpi1(B(2, Srp)) ∼= colimN (2,Srp)A ∼= Ŝrp .2Indeed for the central products D8 ◦D8 and D8 ◦ Q8, that is r = 2, thehigher homotopy groups are given bypii(B(2, S22)) ∼= pii(151∨S2) for i > 1.The point here is to detect torsion elements in the kernel T (2) of the naturalmap ψ : G(2)→ G. This is enough to conclude that the space B(2, G) is nota K(pi, 1) space. We give a group theoretic condition, applicable to the classof symmetric groups Σk and general linear groups GLn(Fq) over a finite fieldof characteristic p. More precisely, the kernel T (2) has a torsion element forsymmetric groups Σk and general linear groups GLn(Fq) when k ≥ 8 andn ≥ 4. In particular for these groups B(2, G) is not a K(pi, 1) space. Tostate the theorem we need a definition. We call a sequence of non-identityelements {gi}2ri=1 a symplectic sequence if the following conditions are satisfied1. [gi, gi+r] 6= 1 for 1 ≤ i ≤ r and [gi, gj] = 1 for any other pair,2. [gi, gi+r] = [gj, gj+r] for all 1 ≤ i, j ≤ r.Theorem. Suppose that G is a finite group which has a symplectic sequence{gi}2ri=1 for some r ≥ 2. Then the kernel T (2) has a torsion element of orderp for each prime dividing the order of c = [gi, gi+r]. Moreover, the spaceB(2, G) is not a K(pi, 1) space.Another natural question is to compute the complex K-theory of a homo-topy colimit. The main tool to study a representable generalized cohomologytheory of a homotopy colimit is the Bousfield-Kan spectral sequence whoseE2-term consists of the derived functors of the inverse limit functor. InChapter 3 we address this problem for the homotopy colimit of classifyingspaces of abelian subgroups of a finite group, that is, for B(2, G). First westudy higher limits of certain functors from the poset dn of proper subsetsof {0, 1, · · · , n} ordered by inclusion to the category Ab of abelian groups.3Theorem. Let R : dn→ Ab be a pre-Mackey functor then lims←R∗ = 0 fors > 0.This theorem can be used to conclude that the E2–term of the spectralsequence consists of torsion groups. The Atiyah–Segal completion theoremalong with this observation implies the following result.Theorem. There is an isomorphismQ⊗Ki(B(2, G)) ∼={Q⊕⊕p||G|Qnpp if i = 0,0 if i = 1,where np is the number of (non-identity) elements of order a power of p inG.Torsion groups can appear in K1(B(2, G)). An explicit example is thecase of S22 , there is an isomorphismKi(B(2, S22)) ∼={Z⊕ Z312 if i = 0,(Z/2)9 if i = 1.In Chapter 4 we give more examples which illustrate the theorems. Anappendix on some properties of simplicial sets is added at the end.4Chapter 1Classifying spaces andhomotopy colimits1.1 Filtrations of classifying spaces1.1.1 PreliminariesLet ∆ be the category whose objects are finite non-empty totally orderedsets n, n ≥ 0, with n+ 1 elements0→ 1→ 2→ · · · → nand whose morphisms θ : m → n are order preserving set maps or alterna-tively functors. The nerve of a small category C is the simplicial setBCn = HomCat(n,C).For instance, the geometric realization of the nerve BG• of a discrete groupG (regarded as a category with one object) is the classifying space BG. An5equivalent way of describing this space is as follows. The assignmentn = (0e1→ 1e2→ · · ·en→ n) 7→ Fnwhere Fn is the free group generated by {e1, e2, ..., en}, defines a faithfulfunctor F : ∆ → Grp injective on objects. Then the classifying space of Gis isomorphic to the simplicial setBG• : ∆op → Setn 7→ HomGrp(Fn, G).Note that this can be regarded as a functor B : Grp→ S via the assignmentG 7→ BG•.A generalization of this construction to certain quotients of free groupsis studied in [1]. We recall the constructions and describe some alternativeversions.Definition 1.1.1. For a groupQ define a chain of groups inductively: Γ1(Q) =Q, Γq+1(Q) = [Γq(Q), Q]. The descending central series of Q is the normalseries1 ⊂ · · · ⊂ Γq+1(Q) ⊂ Γq(Q) ⊂ · · · ⊂ Γ2(Q) ⊂ Γ1(Q) = Q.Now take Q to be the free group Fn. For q > 0, the natural mapsFn → Fn/Γq(Fn) induce inclusions of setsHomGrp(Fn/Γq(Fn), G) ⊂ HomGrp(Fn, G).Furthermore, the simplicial structure of BG• induces a simplicial structureon the collection of these subsets.Definition 1.1.2. Let G be a discrete group. We define a sequence of sim-6plicial sets B•(q,G), q ≥ 2, by the assignmentn 7→ HomGrp(Fn/Γq(Fn), G)and denote the geometric realization |B(q,G)•| byB(q,G). DefineB•(∞, G) =BG• and B(∞, G) = BG by convention.As in [1] this gives rise to a filtration of the classifying space of GB(2, G) ⊂ B(3, G) ⊂ · · · ⊂ B(q,G) ⊂ B(q + 1, G) ⊂ · · · ⊂ B(∞, G) = BG.This construction of a filtration can be generalized to any functor P : ∆ →Grp with a natural transformation λ : F → P , where F is the faithful functorn 7→ Fn introduced at the beginning. Consider the functor BP : Grp→ Topdefined by sending G to the geometric realization of the simplicial set definedby n 7→ Hom(P (n), G), and the natural transformation BP → BF induced byλ. Choosing P (n) as the quotients Fn/Γq(Fn) recovers the spaces B(q,G). Inthe next section we will consider another example of this construction whereP (n) is the pro-p completion Pn of the free group Fn and λ is the completionmap λ : Fn → Pn. One can also work in the category of topological groupsand simplicial spaces. In this case the sets Hom(P (n), G) can be given atopology by the transformation λ induced from Hom(Fn, G) regarded as then–fold product G× · · · ×G︸ ︷︷ ︸n times. A useful property of these functors BP is thatthey preserve finite products.Proposition 1.1.3. There is a natural homeomorphism BP (G1 × G2) →BP (G1)×BP (G2) induced by the projections pii : G1 ×G2 → Gi.Proof. At the simplicial level there are natural mapsHom(P (n), G1 ×G2)→ Hom(P (n), G1)× Hom(P (n), G2)induced by the natural projections. This map is a bijection by the universal7property of products, and compatible with the simplicial structure. Theresult follows from the fact that the geometric realization functor preservesproducts.1.1.2 p-local versionWe discuss an alternative construction which is obtained by replacing thefree group Fn by its pro-p completion, namely the free pro-p-group Pn, see[9] for their basic properties. It can be defined using the p-descending centralseries of the free groupPn = lim←Fn/Γqp(Fn)where Γqp(Fn) are defined recursively:Γ1p(Fn) = Fn and Γq+1p (Fn) = [Γqp(Fn), Fn](Γqp(Fn))p.The group Pn contains Fn as a dense subgroup. Observe that each quotientFn/Γqp(Fn) is a finite p-group.Let TGrp denote the category of topological groups. We consider con-tinuous group homomorphisms φ ∈ HomTGrp(Pn, G) where G is a discretegroup. The image of φ is a finite p-subgroup of G. This follows from the factthat Pn is compact and if K is a subgroup of Pn of finite index then |Pn : K|is a power of p by [9, Lemma 1.18]. Then there exists q ≥ 1 such that φfactors asPnφ> GFn/Γqp(Fn).∨>This implies that HomTGrp(Pn, G) ⊂ HomGrp(Fn, G) and also there is asimplicial structure induced from BG•.We remark that a theorem of Serre on topological groups says that any(abstract) group homomorphism from a finitely generated pro-p group to8a finite group is continuous [9, Theorem 1.17]. This implies that when Gis finite any (abstract) homomorphism φ : Pn → G is continuous. Moregenerally, as a consequence of [22, Theorem 1.13] if every subgroup of G isfinitely generated then the image of an abstract homomorphism φ is a finitegroup, hence in this case φ is also continuous.Definition 1.1.4. Let p be a prime integer and G a discrete group. Wedefine a sequence of simplicial sets B•(q,G)p, q ≥ 2, byn 7→ HomTGrp(Pn/Γq(Pn), G).Define B•(∞, G)p to be the simplicial set n 7→ HomTGrp(Pn, G) and setB(∞, G)p = |B•(∞, G)p|.Let us examine the simplicial set B•(∞, G)p. There is a natural mapθ : lim→qHom(Fn/Γqp(Fn), G)→ Hom(Pn, G)induced by the projections Pn → Fn/Γqp(Fn). This map is injective since itis induced by injective maps and surjective since φ : Pn → G factors throughFn/Γqp(Fn)→ G for some q ≥ 1, as observed above. Therefore θ is a bijectionof sets.Definition 1.1.5. Let p be a prime integer and G a discrete group. Wedefine a sequence of simplicial sets B•(q,G, p), q ≥ 2, byn 7→ HomGrp(Fn/Γqp(Fn), G)and denote the geometric realization |B(q,G, p)•| by B(q,G, p).The facts that colimits in the category of simplicial sets can be con-structed dimension-wise and the geometric realization functor commutes withcolimits imply the following.9Proposition 1.1.6. Suppose that G is a discrete group then θ induces ahomeomorphismlim→qB(q,G, p)→ B(∞, G)p.1.2 The case of finite groupsWe restrict our attention to finite groups. Note that the following collectionsof subgroups have an initial object (the trivial subgroup), and they are closedunder taking subgroups and the conjugation action of G. Define a collectionof subgroups for q ≥ 2N (q,G) = {H ⊂ G| Γq(H) = 1}.These are nilpotent subgroups of G of class less than q. Observe thatN (2, G)is the collection of abelian subgroups of G.Denote the poset of p-subgroups of G by Sp(G) (including the trivialsubgroup) and setN (q,G)p = Sp(G) ∩N (q,G)when q = 2 this is the collection of abelian p-subgroups of G.1.2.1 Homotopy colimitsIn [1] it is observed that there is a homeomorphismcolimN (q,G)BA→ B(q,G)and furthermore the natural maphocolimN (q,G)BA→ colimN (q,G)BA10turns out to be a weak equivalence. A similar statement holds for the spacesB(q,G)p.Proposition 1.2.1. Suppose that G is a finite group. Then there is a home-omorphismB(q,G)p ∼= colimN (q,G)pBPand the natural maphocolimN (q,G)pBP → colimN (q,G)pBPis a weak equivalence.Proof. There is a natural bijection of setscolimN (q,G)pHomGrp(Fn, P )→ HomGrp(Pn/Γq(Pn), G).The image of a morphism Pn/Γq(Pn)→ G is a p-group of nilpotency class lessthan q. Conversely, if P ⊂ G is a p-group of nilpotency class less than q thenFn → P induces a map Pn → P by taking the pro-p completions, and factorsthrough the quotient Pn/Γq(Pn). This induces the desired homeomorphismB(q,G)p ∼= colimN (q,G)pBP . The natural map from the homotopy colimit to theordinary colimit is a weak equivalence since the diagram of spaces is free(Appendix).Remark 1.2.2. For the homotopy colimits considered in this thesis, it issufficient to consider the sub-poset determined by the intersections of themaximal objects of the relevant poset. More precisely, let M1,M2, ...,Mkdenote the maximal groups in the poset N (q,G) andM(q,G) = {∩JMi| J ⊂{1, 2, ..., k}}. The inclusion mapM(q,G)→ N (q,G)11is right cofinal and hence induces a weak equivalence ([28, Proposition 3.10])hocolimM(q,G)BM → hocolimN (q,G)BA.A similar observation holds for N (q,G)p.Proposition 1.2.3. Let Z denote the intersection of the maximal objects inN (q,G). Then there is a commutative diagram of fibrationshocolimM(q,G)G/A hocolimM(q,G)G/AyyBZ −−−→ B(q,G) −−−→ hocolimM(q,G)BA/Z∥∥∥yyBZ −−−→ BG −−−→ BG/Z.Proof. Only the right hand column and the middle row fibrations are unclear.Both follows from an application of the Theorems of Puppe [14, AppendixHL]. The first one is a homotopy colimit of a diagram of fibrations over afixed base and the second is a diagram of fibrations whose fibers are homotopyequivalent to BZ. We prove the first one, the other is similar to the fibrationin the middle column which was discussed before. Note that every object inthe poset contains Z, and the diagramBN −−−→ BMyyBN/Z −−−→ BM/Zinduced by the inclusion BN/Z ⊂ BM/Z is a pull-back diagram which isidentity on the fibre BZ. The fiber of the homotopy colimit is the commonvalue BZ by the Theorem of Puppe.121.2.2 A stable decomposition of B(q,G)Recall that a finite nilpotent group N is a direct product of its Sylow p-subgroupsN ∼=∏p||N |N(p).For a prime p dividing the order of the group G, and N ∈ N (q,G) definepip(N) ={N(p) if p||N |1 otherwise .Lemma 1.2.4. The inclusion map ιp : N (q,G)p → N (q,G) of posets inducesa weak equivalencehocolimN (q,G)pBP → hocolimN (q,G)Bpip(A).Proof. By freeness (Appendix) it is enough to consider ordinary colimits. LetB : N (q,G)p → Top denote the functor P 7→ BP . The map pip : N (q,G)→N (q,G)p defined by A 7→ pip(A) induces an inverse to the map induced by ιpon the colimitcolimN (q,G)pBP → colimN (q,G)Bpip(A).This follows from the fact that pip ◦ ιp and ιp ◦ pip induce the identity onBP → Bpip(P ) = BP and Bpip(A)→ Bιppip(A) = Bpip(A).Theorem 1.2.5. Suppose that G is a finite group. There is a natural weakequivalence∨p||G|ΣB(q,G)p → ΣB(q,G) for all q ≥ 2induced by the inclusions B(q,G)p → B(q,G).Proof. First note that the nerve of the poset N (q,G) is contractible sincethe trivial subgroup is an initial object. Note that for A ∈ N (q,G) each BAis pointed via the inclusion B1→ BA.13Let J denote the pushout category 0← 01→ 1. Define a functor F fromthe product category J ×N (q,G) to Top byF ((0, A)) = pt, F ((01, A)) = BA and F ((1, A)) = pt.Commutativity of homotopy colimits ([5, 4.5.20]) implies thathocolimJhocolimN (q,G)F ∼= hocolimJ×N (q,G)F ∼= hocolimN (q,G)hocolimJF.The nerve of N (q,G) is contractible, in which case we can identify the ho-motopy colimit over J as the suspension and conclude that the mapΣ(hocolimN (q,G)BA)→ hocolimN (q,G)Σ(BA) (1.1)is a homeomorphism. The natural map induced by the suspension of theinclusions BA(p) → BA∨p||A|ΣBA(p) → ΣBAis a weak equivalence. This follows from the splitting of suspension of prod-ucts:Σ(BA(p) ×BA(q)) ' Σ(BA(p)) ∨ Σ(BA(q)) ∨ Σ(BA(p) ∧BA(q))and from the equivalence Σ(BA(p) ∧BA(q)) ' pt when p and q are coprime.The latter follows from the Ku¨nneth theorem and the Hurewicz theoremby considering the homology isomorphism induced by the inclusion BA(p) ∨BA(q) → BA(p) ×BA(q).Invariance of homotopy colimits under natural transformations whichinduce a weak equivalence on each object, the homeomorphism (1.1) and14Lemma 1.2.4 give the following weak equivalencesΣ(hocolimN (q,G)BA) ∼= hocolimN (q,G)Σ(BA)' hocolimN (q,G)∨p||A|Σ(BA(p))'∨p||G|Σ(hocolimN (q,G)Bpip(A))'∨p||G|Σ(hocolimN (q,G)pBP ).When commuting the wedge product with the homotopy colimit, again anargument using the commutativity of homotopy colimits can be used similarto the suspension case. Note that the wedge product is a homotopy colimit.This stable equivalence immediately implies the following decompositionfor a generalized cohomology theory.Theorem 1.2.6. There is an isomorphismh˜∗(B(q,G)) ∼=∏p||G|h˜∗(B(q,G)p) for all q ≥ 2where h˜∗ denotes a reduced cohomology theory.Therefore one can study homological properties of B(q,G) at a fixedprime. In particular, this theorem applies to the complexK-theory of B(q,G)and each piece corresponding toB(q,G)p can be computed using the Bousfield-Kan spectral sequence [6].1.2.3 Homotopy types of B(q,G) and B(q,G)pWe follow the discussion on the fundamental group ofB(q,G) from [1], similarproperties are satisfied by B(q,G)p. Let N be a collection of subgroups of a15finite group G. We require that N is closed under taking subgroups. Notethat the trivial group is the initial object. In particular, it can be takenas N (q,G) or N (q,G)p. The fundamental group of a homotopy colimit isdescribed in [13, Corollary 5.1]. The trivial group is the initial object in Nand its classifying space B1 is regarded as a base point of the spaces in thediagram. Therefore there is an isomorphismpi1(hocolimNBA) ∼= colimNAwhere the colimit is in the category of groups. We will study colimits of(abelian) groups in the next section in more detail.For each A ∈ N , there is a natural fibration G/A → BA → BG andtaking homotopy colimits one obtains a fibrationhocolimNG/A→ hocolimNBA→ BG. (1.2)This is a consequence of a Theorem of Puppe, see [14, Appendix HL]. Thepoint is that each fibration has the same base space BG. Associated to thisfibration there is an exact sequence of homotopy groups1→ pi1(hocolimNG/A)→ pi1(hocolimNBA)ψ→ pi1(BG)→ pi0(hocolimNG/A)→ 0 (1.3)where ψ is the natural map pi1(hocolimNBA) ∼= colimNA→ G induced by theinclusions A→ G since the diagramhocolimNBA > BGBA∧>commutes for all A ∈ N . Note that if N contains all cyclic subgroups of G16then hocolimNG/A is connected. Since in this case, the commutativity ofcolimNAψ> G〈g〉∧>implies that ψ is surjective.We point out here that for N = N (q,G) the homotopy fibre hocolimNG/Ais homotopy equivalent to the pull-back of the universal principal G-bundleE(q,G) −−−→ EGyyB(q,G) −−−→ BGand this pull-back E(q,G) can be defined as the geometric realization of asimplicial set as described in [1]. It can be identified as a colimitE(q,G) ∼= colimN (q,G)G×A EAand there is a commutative diagramhocolimN (q,G)G×A EA∼−−−→ hocolimN (q,G)G/Ay∼ycolimN (q,G)G×A EA −−−→ colimN (q,G)G/Ainduced by the contractions EA → pt, and the weak equivalences as indi-cated. The exact sequence (1.3) becomes0→ T (q)→ G(q)ψ→ G→ 0, (1.4)where G(q) = pi1(B(q,G)) and T (q) = pi1(E(q,G)).17Consider the universal cover B˜(q,G) of B(q,G). Again using the Theoremof Puppe, it can be described as the homotopy fibre of the homotopy colimitof the natural fibrations G(q)/A→ BA→ BG(q) ([1, Theorem 4.4])B˜(q,G) ' hocolimN (q,G)G(q)/A −−−→ B(q,G)yBG(q).The question of B(q,G) having the homotopy type of a K(pi, 1) space isequivalent to asking whether the classifying space functor B commutes withcolimitsB(q,G) = colimN (q,G)BA→ B(colimN (q,G)A).The difference is measured by the simply connected, finite dimensional com-plex B˜(q,G), or equivalently by the values of the higher limitsH i(B˜(q,G);Z) ∼= limi←N (q,G)Z[G(q)/A]as implied by the Bousfield-Kan spectral sequence.18Chapter 2Finiteness of G(2) andhomotopy type of B(2, G)The fundamental groups of B(2, G) and B(2, G)p are isomorphic to the col-imits of the collection of abelian subgroups and abelian p-subgroups of G,respectively. Therefore it is natural to study the colimit of a collection ofabelian groups in greater generality to determine the homotopy properties ofthese spaces. Then we turn to a closer study of the colimit G(2) of abeliansubgroups. We give a group theoretic condition on G which implies the exis-tence of torsion elements in the kernel T (2) of the natural map ψ : G(2)→ G.This allows us to conclude that B(2, G) is not a K(pi, 1) space for the groupssatisfying this condition. Extraspecial p–groups of rank r ≥ 4 are such ex-amples. For further examples see Chapter 4.2.1 Colimits of solvable groupsLet A denote a collection of solvable finite groups closed under taking sub-groups. We consider the colimit of the groups in A. This can be constructed19as a quotient of the free product∐of the groups in the collectioncolimAA ∼= (∐A∈AA)/ ∼by the normal subgroup generated by the relations b ∼ f(b) where b ∈ Band f : B → A runs over the morphisms in A. We need a definition fromgroup theory.Definition 2.1.1. A chief series of a group G is a series of normal subgroups1 = N0 ⊂ N1 ⊂ ... ⊂ Nk = Gfor which each factor Ni+1/Ni is a minimal (nontrivial) normal subgroup ofG/Ni. Let d(G) denote the number of indices i = 1, 2, ..., k such that Ni/Ni−1has a complement in G/Ni−1For r > 0 define a sequence of sub-collections Ar = {A ∈ A| d(A) ≤ r}.The main result of this section is the following isomorphism of colimits, whichreduces the collection A to the sub-collection A2.Theorem 2.1.2. Let A be a collection of solvable finite groups closed undertaking subgroups. The natural mapcolimA2A→ colimAAinduced by the inclusion map A2 → A is an isomorphism.The proof follows from the homotopy properties of a complex constructedfrom the cosets of proper subgroups of a group. We review some backgroundfirst.The coset posetConsider the poset {xH| H ( G} consisting of cosets of proper subgroups ofG ordered by inclusion. The associated complex is denoted by C(G) and it20is studied in [7]. This complex can be identified as a homotopy colimitC(G) ' hocolimH∈P(G)G/H,where P(G) = {H ( G}. The identification follows from the description ofthis homotopy colimit as the nerve of the transport category of the poset{H ( G} which is precisely the nerve of the poset {xH| H ( G}.When G is solvable, C(G) has the homotopy type of a bouquet of spheresof dimension d(G).Proposition 2.1.3. [7, Proposition 11] Suppose that G is a solvable finitegroup and1 = N0 ⊂ N1 ⊂ ... ⊂ Nk = Gbe a chief series thenC(G) 'n∨Sd(G)−1for some n > 0, where d(G) is the number of indices i = 1, 2, ..., k such thatNi/Ni−1 has a complement in G/Ni−1.We are interested only in the dimensions of the spheres. An immediateconsequence of this result is the following.Corollary 2.1.4. Let G be a finite solvable group such that d(G) ≥ 3, andP(G) denote the poset of proper subgroups {H ( G}. Then the natural mapβ : colimP(G)A→ Gis an isomorphism.Proof. Observe that the fiber of the natural map hocolimP(G)BA → BG isthe coset poset C(G) ' hocolimP(G)G/A, and it is homotopy equivalent to∨n Sd(G)−1 by Proposition 2.1.3. The result follows immediately from theassociated exact sequence of homotopy groups.21Proof of Theorem 2.1.2. We have a sequence of inclusionsA2 → A3 → ...→ Ai → Ai+1 → ...→ Awhich can be filtered further as followsAi = Ai,0 ⊂ Ai,1 ⊂ ... ⊂ Ai,j ⊂ ... ⊂ Ai+1where Ai,j = {B ∈ Ai+1| P(B) ⊆ Ai,j−1} for j > 0. For i ≥ 2 and j ≥ 0,each of the mapsΛ : colimAi,jA→ colimAi,j+1Ahas an inverse induced by the following compositions: Let Q ∈ Ai,j+1Qβ−1→ colimP¯(Q)∩Ai,jAθ→ colimAi,jAwhere P¯(Q) is the poset of (all) subgroups of Q, β is the map in Corollary2.1.4, and the map θ is induced by the inclusion P¯(Q)∩Ai,j → Ai,j. This isa consequence of the commutativity ofcolimAi,jAΛ−−−→ colimAi,j+1AxθxιQcolimP¯(Q)∩Ai,jAβ−−−→∼=Q.Hence each inclusion map Ai,j → Ai,j+1 induces an isomorphism on thecolimits.2.2 Elementary properties of G(2)Recall that the groups G(q) are defined to be pi1(B(q,G)) and comes with anatural map ψ : G(q)→ G. For a fixed q it is described as the colimit of the22collection N (q,G) of nilpotent subgroups of class less than q. The reductiontheorem 2.1.2 simplifies this description.Proposition 2.2.1. G(q) is isomorphic to the colimit of the groups in thesub-collection N2 ⊂ N (q,G) consisting of p–groups of rank at most 2 andproducts P × Q of a p–group with a q–group both of rank equal to 1, wherep and q are distinct primes. Here the rank of a p–group is the rank of itsFrattini quotient.Proof. By Theorem 2.1.2, the colimit is determined by the subgroups N suchthat d(N) ≤ 2. Since N is nilpotent it is isomorphic to the product∏N(p)of its Sylow p–subgroups. Note that for a p–group P the number d(P ) is atleast the rank of the group, since the Frattini quotient P  P/Φ(P ) can berefined into a chief series.Also one can give a presentation of G(2):G(2) = 〈(g)| g ∈ G, (g)(h) = (gh) if [g, h] = 1 in G〉 (2.1)as a quotient of the free group F (G) generated on the set of elements of G.To see this, note that the homomorphism F (G)→ G(2) induced by sending(g) to the image of g under the natural map 〈g〉 → G(2) factors through therelations given in the presentation. There is also a map G(2) → F (G)/Rinduced by the inclusions A → F (G)/R of abelian subgroups A of G whereR is the normal subgroup of F (G) generated by the relations given in thepresentation. These two maps are inverses of each other. In this presentationthe natural map G(2) → G corresponds to (g) 7→ g. By adding furtherrelations to this presentation G(q) can be described as a quotient of the freegroup as well. Then there is a sequence of surjective group homomorphismsG(2)→ · · · → G(q − 1)→ G(q)→ · · · → G(∞) = G.Proposition 2.2.1 suggests that the group G(q) is closely related to the23free product of the groups G(q)p = pi1(B(q,G)p) which is a colimit over thecollection N (q,G)p of p–subgroups of nilpotency class less than q. There is anatural map G(q)p → G(q) induced by the inclusion N (q,G)p ⊂ N (q,G) foreach prime p which divides the order of the group. Furthermore 2.2.1 impliesthat the kernel of the map∐p||G|G(q)p → G(q)is generated by the commutators of coprime order subgroups in G, hence iscontained in the kernel of the natural map∐G(q)p →∏G(q)p. Thereforethere exist a map G(q) →∏G(q)p which splits via the natural maps asshown in the following diagram∐pG(q)p −−−→ G(q)xyG(q)p ←−−−∏G(q)p.(2.2)Proposition 2.2.2. The natural map G(q)p → G(q) is an inclusion, and itsplits.Proof. The splitting is given by the composition of the map G(q)→∏G(q)pin 2.2 with the natural projection∏G(q)p → G(q)p.There is a nicer interpretation of the splitting in the case of G(2). Firstobserve that sending an element to its n-th power defines a map of spacesωn : B(2, G)→ B(2, G)at the simplicial level. This is induced by the map (g1, · · · , gk) 7→ (gn1 , · · · , gnk )which is well–defined and respects the simplicial structure maps since gicommutes with each other. The induced map on the fundamental groupswill also be denoted by the same symbol. Writing the order |G| of the group24as a product of distinct primes∏pnii and defining qi = |G|/pnii , one can seethat the image ofωqi : G(2)→ G(2)is isomorphic to G(2)pi . Moreover, the composition ωq′i ◦ωqi , where q′i ≡ 1/qimod pnii , restricted to G(2)pi gives a splitting of the natural inclusion mapG(2)pi → G(2). There are other useful properties one can deduce using themaps ωn which we discuss next.A natural question is to ask when G(2) is equal to G. This can beanswered by considering the automorphism ω−1 of G(2) of order 2.Proposition 2.2.3. G is an abelian group if and only if the natural mapψ : G(2)→ G is an isomorphism.Proof. Consider the diagramG(2)(−1)−−−→ G(2)yψyψG G.If ψ is an isomorphism then the composition ψ ◦ (−1) ◦ ψ−1 : G → G is anisomorphism, that is the inversion map g 7→ g−1 is an isomorphism. Then Gis abelian. The converse is clear from the definition of G(2).Corollary 2.2.4. The natural map B(2, G)→ BG is a homotopy equivalenceif and only if G is an abelian group.2.3 Finiteness of G(2)We denote the commutator subgroup by G′ and the quotient G/G′ by Gab.In this section we discuss certain class of (non-abelian) finite groups for whichG(2) is a finite group. First we introduce a finite group which is a sort of25intermediary between G(2) and G. Consider the group homomorphismψ̂ : G(2)→ G×G, (g) 7→ (g, g−1)and denote its image by Ĝ. Clearly, this image is in the kernel ofm¯ : G×G→ Gab, (g1, g2) 7→ g1g2G′,and we claim that Ĝ = ker(m¯). First note that as a set ker(m¯) is the disjointunion∐gG′⊂G gG′×g−1G′ hence has order |G||G′|, whereas Ĝ is isomorphic tothe quotient of G(2) by the intersection I = ker(ψ)∩ω−1(ker(ψ)). Thereforeit suffices to show that the index | ker(ψ) : I| is equal to |G′|, which is alsoequal to the index |ω−1(ker(ψ)) : I|. Observe that if [g1, g2] 6= 1 in G then(g−11 )(g−12 )(g2g1) is in the kernel of ψ but not contained in I. Therefore(g1)(g2)(g2g1)−1 is in ω−1(ker(ψ)) and the subgroup generated by the imagesof such elements under ψ̂ contains G′. This proves the claim. Moreover thereis a commutative diagramG(2)ψ̂> ĜG,pi1∨ψ >(2.3)where pi1 is induced by projecting onto the first coordinate.Proposition 2.3.1. The commutator subgroup (Ĝ)′ of Ĝ is isomorphic tothe pull–back ofG′yG′ −−−→ G′/[G,G′]under the natural projections. Moreover, G′ ⊂ Z(G) if and only if (Ĝ)′ isthe diagonal subgroup ∆(G′) = {(g, g)| g ∈ G′}.Proof. The commutator (Ĝ)′ is a characteristic subgroup of Ĝ, hence normal26in G×G. Therefore any commutator of the form [(t, 1), [(g, g−1), (h, h−1)]] =([t, [g, h]], 1) lies in (Ĝ)′ for all t, g, h ∈ G. Furthermore, the group Ĝ isinvariant under the swap map (x, y) 7→ (y, x) which implies that the product[G,G′]× [G,G′] is contained in (Ĝ)′. There is a commutative diagram[G,G′]× [G,G′] −−−→ (Ĝ)′ −−−→ (Q̂)′∥∥∥yy[G,G′]× [G,G′] −−−→ Ĝ −−−→ Q̂where Q denotes the quotient G/[G,G′]. Note that it suffices to prove thelast part of the proposition for Q, which satisfies Q′ ⊂ Z(Q). Recall thatQ̂ is generated by (g, g−1), g ∈ Q, in Q × Q. Then for all g, h ∈ Q wehave [(g, g−1), (h, h−1)] = ([g, h], [g−1, h−1]) = ([g, h], [g, h]) which proves onedirection. For the converse, assume that (Q̂)′ = ∆(Q′). The commutator(Q̂)′ is normal in Q × Q, i.e. (1, g)(g′, g′)(1, g−1) ∈ ∆(Q′) for all g ∈ Q andg′ ∈ Q′.Proposition 2.3.2. Let θ : G → H be a surjective (injective) group homo-morphism. Then there is a commutative diagramker(ψ̂G) −−−→ Ĝ −−−→ Gyyyker(ψ̂H) −−−→ Ĥ −−−→ H,where all the vertical maps are surjective (injective).Proof. Recall that Ĝ is the kernel of G×G→ G/G′ the composition of theabelianization map with the multiplication map. Therefore the conclusionof the proposition holds for the middle vertical arrow. The kernel ker ψ̂G isthe product 1 × G′ since ψ̂G is the projection pi1 onto the first factor. Thefirst vertical arrow between the kernels is the map between the commutators1×G′ → 1×H ′, hence it is surjective (injective) if the initial map G → H27is surjective (injective).In the next sections we will consider certain classes of p–groups whereψ̂ turns out to be an isomorphism. The following observation justifies ourrestriction to p–groups.Proposition 2.3.3. Assume G is a finite group. If G(2) is a finite groupthen G is nilpotent, i.e. isomorphic to the product of its Sylow p–subgroups.Proof. Assume that G(2) is finite, we will prove that G is nilpotent. Noteby 2.2.3 each G(2)p, p||G|, is also finite. If G is a p–group then there isnothing to prove. So assume G has composite order. In this case we willshow G is isomorphic to the product of its Sylow p–subgroups. In otherwords, any two elements of coprime order commutes. Let p and q be distinctprimes which divide the order of G. Assume the contrary that there arenon–trivial elements g and h in G of order a power of p and q respectivelysuch that [g, h] 6= 1. The product (g)(h) has infinite order in the free product∐p||G|G(2)p since it is cyclically reduced ([25, 1.3, Proposition 2]), where (g)and (h) denotes the corresponding generators in G(2)p and G(2)q. Assumethat its order in G(2) is n. From 2.2 we obtain the diagramG(2)p∐G(2)q > G(2)G(2)p∏G(2)q,∨α>where (g)(h) maps to ((g), (h)) in the product. Therefore the order n ofthe image of (g)(h) in G(2) is divisible by the order p of g and the orderq of h. Now the product ((g)(h))n can be written in G(2) as a product ofcommutators as follows(g)(h) · · · (g)(h)︸ ︷︷ ︸n= [(g), (h)][(h), (g)2] · · · [(h)n−1, (g)n](g)n(h)n= [(g), (h)][(h), (g)2] · · · [(h)n−1(g)n]28which is assumed to be in the kernel of G(2)p∐G(2)q → G(2). This kernelis also a subgroup of the kernel of α which is a free group whose basis con-sists of the commutators [x, y] of elements in G(2)p and G(2)q, see [25, 1.3,Proposition 4]. Therefore there is a consecutive pair of commutators whichare inverses of each other. This implies [(g), (h)] = 1, hence [g, h] = 1, whichis a contradiction.2.3.1 Extraspecial p–groupsWe will restrict attention to central extensions of the formZ/p P pi→ Vwhere V is an elementary abelian p–group, which can also be regarded asa vector space over Z/p. Let ρ denote a representative of the class of theextension. There is a bilinear form associated to ρ defined by b = ρ + ρtwhere ρt(x, y) = ρ(y, x). In this case b : V × V → Z/p is given by b(v, w) =[pi−1(v), pi−1(w)]. In particular, b is skew symmetric b(v, w) = −b(v, w). Wefurther assume that this bilinear form is non-degenerate. This is equivalentto having Z(P ) = Z/p. Hence by standard results [12, Chapter I] in thetheory of bilinear forms (V, b) is isometric to an orthogonal sum rH−1 of rcopies of the hyperbolic plane. In particular the dimension of V is even,dim(V ) = 2r. A group of this type is called an extraspecial p-group. At afixed r there are two types up to isomorphism. One can choose a symplecticbasis {ei}1≤i≤2r where b(ei, ei+r) = 1 and for any other pair b(ei, ej) = 0.Definition 2.3.4. Let b be a bilinear form on a vector space V over a fieldof prime characteristic and I(b) denote the collection of isotropic subspacesW , i.e. the restriction b|W is the zero form. Define the following groupV (b) = colimW∈IW29as a colimit in the category of groups. Here W is regarded as an elementaryabelian p–group.Note that by Proposition 1.2.3 there is a central extensionZ/p ↪→ P (2)  V (b),which will be used in the next result. A presentation of V (b) similar to 2.1can be given as〈(v) ∈ V | (v)(w) = (v + w) if b(v, w) = 0〉.The subgroup Sp(V, b) of linear automorphisms Aut(V ) which leaves b in-variant, i.e. α ∈ GL(V ) such that b(α(v), α(w)) = b(v, w), acts on the groupV (b) via the automorphism (v) 7→ (α(v)).Theorem 2.3.5. Let P be an extraspecial p-group. Then the map ψ̂ : P (2)→P̂ in 2.3 is an isomorphism if and only if r ≥ 2.Proof. When r = 1 the group V (b) is the free product of its cyclic subgroupshence P (2) is infinite. We assume r ≥ 2, and observe that the conclusion ofthe theorem holds for P if and only if V (b) has order |P |. Choose a symplecticbasis {ei}2ri=1 for V . Given i 6= j in {1, · · · , r} the following equations holdin V (b)(ei)(ei+r)(ei)−1(ei+r)−1 = (ei)(ej − ej)(ei+r)(ei)−1(ej+r − ej+r)(ei+r)−1= (ej)(ei)(−ej + ei+r)(−ei + ej+r)(ei+r)−1(−ej+r)= (ej)(ei)(−ei + ej+r)(−ej + ei+r)(ei+r)−1(−ej+r)= (ej)(ej+r)(ej)−1(ej+r)−1,where we make use of the identity (v)−1 = (−v). The key step is the obser-vation that b(−ej +ei+r,−ei+ej+r) = −b(−ei, ei+r)+b(−ej, ej+r) = 0. Thisshows, in particular, that the commutator [(ei), (ei+r)] is central. Here we30use the fact that an element (v) for some v ∈ V can be written as a product∏ri=1(αiei + βiei+r) in V (b).Now any pair v, w ∈ V with b(v, w) 6= 0 can be mapped to the hyperbolicpair ei, ei+r after renormalizing w to w¯ such that b(v, w¯) = 1 by sending v, w¯to ei, ei+r. By Witt’s Lemma [3, pg. 81] the isometry 〈v, w¯〉 → 〈ei, ei+r〉extends to a map α ∈ Sp(V, b). Hence [(v), (w¯)] is also central as well as thecommutator [(v), (w)] since (w) = (b(v, w)w¯). We see that the commutatorof V (b) is cyclic of order p and its abelianization is an elementary abelianp–group of rank 2r. This proves that V (b) has the same order as P .Observe that Proposition 2.3.1 can be used to prove that P̂ sits in acentral extensionZ/p P̂ → (Z/p)2r−1 × T if p odd,where T ∼= Z/p2 if the exponent of P is p2, and T ∼= (Z/p)2 otherwise, andfor p = 2Z/2 P̂ → (Z/2)2r+1.We can extend our consideration to an arbitrary b not necessarily non–degenerate without difficulty. Writing b = b′ ⊥ 0|V ⊥ where b′ is non–degenerate and V ⊥ denotes the radical of b, one has V (b) ∼= V (b′)× V ⊥.Corollary 2.3.6. Let P be a (non–abelian) group which fits in a centralextension Z/p  P → V where V is an elementary abelian p–group. ThenP (2) ∼= P̂ if and only if dim(b′) ≥ 4, where b is the bilinear form associatedto the commutator and b′ is its non–degenerate part.2.4 Homotopy type of B(2, G)In this section we generalize some ideas used in Theorem 2.3.5 to detecttorsion in the kernel T (2) of the natural map ψ : G(2)→ G. This enables us31to decide whether B(2, G) is a K(pi, 1) space by the following result.Proposition 2.4.1. If the natural map B(q,G) → BG(q) is a homotopyequivalence then the kernel T (q) of the natural map G(q) → G is torsionfree.Proof. Recall that the fiber of the map B(q,G) → BG is a finite dimen-sional complex. If a finite dimensional complex is a K(pi, 1) space then itsfundmental group is torsion free. This follows from the fact that the coho-mological dimension of pi is less then or equal to its geometric dimension,and pi is torsion free if its cohomological dimension is finite, see [8, ChapterVIII].Definition 2.4.2. Let G be a finite group. We call a sequence of non-identityelements {gi}2ri=1 a symplectic sequence if the following conditions are satisfied1. [gi, gi+r] 6= 1 for 1 ≤ i ≤ r and [gi, gj] = 1 for any other pair,2. [gi, gi+r] = [gj, gj+r] for all 1 ≤ i, j ≤ r.The pair (gi, gi+r) is called a symplectic pair.Denote the subgroup of G generated by the elements {gi}2ri=1 by S. Animmediate observation, which follows from commutator identities, is that theelement c = [gi, gi+r] is central in S, and its order n divides the orders of gifor all i. Moreover the derived subgroup S ′ is cyclic of order n and there isa central extensionZ/n→ S → A,where A is an abelian group of rank 2r generated by the images g¯i. Thebilinear form b associated to the extension class ρ ∈ H2(A;Z/n) is non-zeroon the images of the symplectic pairs b(g¯i, g¯i+r) 6= 0. Let p be a prime divid-ing n, and Srp denote an extraspecial p-group of rank 2r with a symplecticsequence {ei}2ri=1, one can take a lift of a symplectic basis in the central quo-tient. We claim that the map S → Srp defined by gi 7→ ei is a surjective group32homomorphism. First note that there is a surjective homomorphism S → Spwhere Sp is the central quotient of S by the subgroup of index p in Z/n.Hence it sits in a central extension Z/p → Sp → A. The associated bilinearform is the composition of b with the projection Z/n → Z/p. With respectto this bilinear form the symplectic sequence {gi} in S maps to a symplecticsequence in Sp. We still denote this sequence by {gi}, and its image in A by{g¯i}. This means that the restriction of the extension class ρ ∈ H2(A;Z/p)corresponding to Sp to the subgroup generated by the image of a symplecticpair (gi, gi+r) in A is non-zero. Therefore, p divides the order of gi for each1 ≤ i ≤ 2r. Choose a surjective homomorphism pi : A → V = (Z/p)2r. Theinduced map on pi∗ : H2(V ;Z/p) → H2(A;Z/p) is also surjective since thep–rank of A is the same as the p–rank of V . Then there exists a class ρVmapping to ρ. The extension corresponding to ρV is an extraspecial p-groupof rank 2r, that is an extension Z/p → Srp → V . Moreover, there is anisomorphism between the pull–back E in the diagramE −−−→ Ay piySrp −−−→ Vand Sp since they have the same extension class. Hence we proved the fol-lowingProposition 2.4.3. Let {gi}2ri=1 be a symplectic sequence in G, and S de-note the subgroup generated by the elements gi. Then there is a surjectivehomomorphism pip : S → Srp which factors through the central quotient Spfor each prime p dividing the order of c = [gi, gi+r]. The induced map on thequotient pi : A→ V sends the image of the symplectic sequence {g¯i}2ri=1 ontoa symplectic basis {ei}2ri=1 of V where ei = pi(g¯i).Lemma 2.4.4. Suppose that {gi}2ri=1 is a symplectic sequence in G and r ≥ 2.Then the equation [(gi), (gi+r)] = [(gj), (gj+r)] holds in G(2) for all 1 ≤ i, j ≤33r. Moreover, the element t = [(gi), (gi+r)]([gi, gi+r])−1 has finite order inG(2).Proof. The proof is mainly the same as in Theorem 2.3.5, a direct compu-tation using the presentation of G(2). For 1 ≤ i 6= j ≤ r the followingequations hold in G(2)(gi)(gi+r)(gi)−1(gi+r)−1 = (gi)(gjg−1j )(gi+r)(gi)−1(gj+rg−1j+r)(gi+r)−1= (gj)(gi)(g−1j gi+r)(g−1i gj+r)(gi+r)−1(g−1j+r)= (gj)(gi)(g−1i gj+r)(g−1j gi+r)(gi+r)−1(g−1j+r)= (gj)(gj+r)(gj)−1(gj+r)−1,where we make use of the identity (g)−1 = (g−1). The key step is the observa-tion that [g−1j gi+r, g−1i gj+r] = [g−1j , gj+r][gi+r, g−1i ] = [gj, gj+r]−1[gi, gi+r] = 1.This shows, in particular, that the element c = [(gi), (gi+r)] commutes with([gi, gi+r])−1. Hence tm = 1 for large enough m.Theorem 2.4.5. Suppose that G is a finite group which has a symplecticsequence {gi}2ri=1 for some r ≥ 2. Then the kernel T (2) of the natural mapψ : G(2) → G has a torsion element of order p for each prime dividing theorder of c = [gi, gi+r]. Moreover, the space B(2, G) is not a K(pi, 1) space,that is the natural map B(2, G)→ BG(2) is not a homotopy equivalence.Proof. By Proposition 2.4.1 it suffices to show that the kernel T (2) of ψ :G(2)→ G has a torsion element. By Lemma 2.4.4 the elementt = [(gi), (gi+r)]([gi, gi+r])−1 ∈ T (2)is a torsion element, it suffices to show that t 6= 1. This will follow from the34commutative diagramSrp <<pip S ⊂ι> GŜrp∧∧<< Ŝ∧∧⊂ιˆ> Ĝ∧∧Srp(2)∼=∧<< S(2)ψ̂S∧∧> G(2)ψ̂G∧∧where Proposition 2.3.2 implies the surjectivity (injectivity) of the relevantmaps. Regarding gi ∈ G as an element of S we see that there is an elementtS in S(2) which maps to t. Now the lower part of the diagram givestp ←−−− tˆS −−−→ tˆ∥∥∥xψ̂Sxψ̂Gtp ←−−− tS −−−→ twhere tp 6= 1 since it generates the kernel of Ŝrp → Srp . Then tˆS 6= 1 whichimplies tˆ 6= 1 since ιˆ is injective. Therefore we obtain the desired result thatt 6= 1, and also p divides the order of t.Corollary 2.4.6. If G has a subgroup isomorphic to Srp for some r ≥ 2 thenB(2, G) is not a K(pi, 1) space.Remark 2.4.7. Note that having a subgroup generated by a symplectic se-quence is stronger than just having a sub–quotient isomorphic to Srp. Con-sider the group D8 × D8 which surjects onto a S22 but B(2, D8 × D8) ∼=B(2, D8)× B(2, D8) by Proposition 1.1.3, and it is a K(pi, 1). The quotientmust be a quotient of a subgroup S generated by a symplectic sequence in Gi.e.Srp  S ↪→ G where r ≥ 2.35Chapter 3Complex K–theory3.1 Higher limitsAs the higher limits appear in the E2-term of the Bousfield-Kan spectral se-quence, in this section we discuss some of their properties. The main theoremof this section is a vanishing result of the higher limits of the contravariantpart of a pre-Mackey functor R : dn → Ab where dn denotes the poset ofnon-empty subsets of {1, ..., n} ordered by reverse inclusion.Let C be a small category and F : C → Ab be a contravariant functorfrom C to the category of abelian groups. AbC denotes the category ofcontravariant functors C→ Ab. Observe thatlim←F ∼= HomAb(Z, lim←F ) ∼= HomAbC(Z, F ).Definition 3.1.1. Derived functors of the inverse limit of F : C → Ab aredefined bylimi←F ≡ ExtiAbC(Z, F ).A projective resolution of Z in AbC can be obtained in the following way,for details see [27]. First note that the functors Fc : C → Ab defined by36Fc(c′) = Z HomC(c′, c) are projective functors and by Yoneda’s lemmaHomAbC(Fc, F ) ∼= F (c).Let C \ − : C → S denote the functor which sends an object c of C tothe nerve of the under category C \ c. The nerve B(C \ c) is contractiblesince the object c → c is initial. Then we define a resolution P∗ → Z of theconstant functor as the composition P∗ = C∗ ◦C \ − where C∗ : S→ Ab isthe functor which sends a simplicial set X to the associated chain complexC∗(X) = Z[X∗]. At each degree n, there are isomorphismsHomAbC(Pn, F ) ∼= HomAbC(⊕x0←···←xnFxn , F ) ∼=∏x0←x1←···←xnF (xn).Differentials are induced by the simplicial maps between chains of morphisms.The following result describes them explicitly.Lemma 3.1.2 ([23]). limi←F ∼= H i(C∗(C;F ), δ) whereCn(C;F ) =∏x0←x1←···←xnF (xn)for all n ≥ 0 and where for U ∈ Cn(C;F )δ(U)(x0 ← x1 ← · · · ← xnφ← xn+1) =n∑i=0(−1)iU(x0 ← · · · ← xˆi ← · · · ← xn+1)+(−1)n+1F (φ)(U(x0 ← · · · ← xn)).A useful property of higher limits, which allows the change of the indexingcategory, is the following.Proposition 3.1.3. [19, Lemma 3.1] Fix a small category C and a con-travariant functor F : C → Ab. Let D be a small category and assume37that g : D → C has a left adjoint. Set g∗F = F ◦ g : D → Ab thenH∗(C;F ) ∼= H∗(D; g∗F ).Assume that the indexing category C is a finite partially ordered set(poset) whose morphisms are i→ j whenever i ≤ j. We consider an appro-priate filtration of F as in [17].Definition 3.1.4. A height function on a poset is a strictly increasing mapof posets ht : C→ Z.A convenient height function can be defined by ht(A) = −dim(|C≥A|)where C≥A is the subposet of C consisting of elements C ≥ A. Let N =dim(|C|) then it is immediate that limi←F vanishes for i > N . The functorF can be filtered in such a way that the associated quotient functors areconcentrated at a single height. Define a sequence of subfunctorsFN ⊂ · · · ⊂ F2 ⊂ F1 ⊂ F0by Fi(A) = 0 if ht(A) > −i and Fi(A) = F (A) otherwise. This induces adecreasing filtration on the cochain complexes hence there is an associatedspectral sequence whose E0-term isEi,j0 = Ci+j(C;Fi/Fi+1) =∏A∈C|ht(A)=−iHomZ(Ci+j(|C≥A|, |C>A|), F (A))with differentials d0 : Ck(C;Fi/Fi+1) → Ck+1(C;Fi/Fi+1). E1-term is thecohomology of the pair (|C≥A|, |C>A|) in coefficients F (A) that isEi,j1 =∏A∈C|ht(A)=−iH i+j((|C≥A|, |C>A|);F (A)). (3.1)Fix A and let A′ > A such that ht(A′) = ht(A) + 1, then differential d1 can38be described as the compositionHk((|C≥A′ |, |C>A′|);F (A′))F (A≤A′)→ Hk((|C>A|, |C>A,ht≥ht(A)+2|);F (A))∂→ Hk+1((|C≥A|, |C>A|);F (A))where ∂ is the boundary map associated to the triple(|C≥A|, |C>A|, |C>A,ht≥ht(A)+2|).We now consider higher limits over specific kinds of diagrams. Let dnbe the category associated to the non-degenerate simplexes of the standardn-simplex, the objects are increasing sequences of numbers σk = [i0 < · · · <ik] where 0 ≤ ij ≤ n and the morphisms are generated by the face mapsdj([i0 < · · · < ik]) = [i0 < · · · < iˆj < · · · < ik] for 0 ≤ j ≤ k.Proposition 3.1.5. For any contravariant functor F : dn→ Ab the spectralsequence (3.1) collapses onto the horizontal axis hence gives a long exactsequence0→∏σ0F (σ0)→∏σ1F (σ1)→ · · · →∏σn−1F (σn−1)→ F (σn)→ 0 (3.2)and where for U ∈ Ck−1dn (F ) =∏σk−1F (σk−1),δk−1(U)(σk) =k∑j=0(−1)k−jF (dj)U(dj(σk)).Proof. For a simplex σk ∈ dn, let dn≥k be the poset of simplices σ ≥ σkand dn>k denote the poset of simplices σ > σk. Observe that the pair(|dn≥k|, |dn>k|) is homeomorphic to (|∆k|, |∂∆k|) where ∆k is the standard39k-simplex with boundary ∂∆k. The spectral sequence of the filtration hasEk,j1 =∏σkHk+j((|dn≥k|, |dn>k|);F (σk)).Therefore E1-term vanishes unless j = 0, and otherwiseEk,01 =∏σkHk((|dn≥k|, |dn>k|);F (σk)) =∏σkF (σk)hence collapses onto the horizontal axes, resulting in a long exact sequence.The differential is induced by the alternating sum of the face maps dj(σk) for0 ≥ j ≥ k.In some cases, it is possible to show that the higher limits vanish. Thenext theorem is an illustration of this instance. First let us recall the defini-tion of a pre-Mackey functor in the sense of Dress [10]. Let M : C→ D be abifunctor, that is a pair of functors (M∗,M∗) such that M∗ is contravariant,M∗ is covariant and both coincide on objects.Definition 3.1.6. A pre-Mackey functor is a bifunctor M : C → D suchthat for any pull-back diagramX1β1−−−→ X2α1y β2yX3α2−−−→ X4in C the diagramM(X1)M∗(β1)−−−−→ M(X2)M∗(α1)x M∗(β2)xM(X3)M∗(α2)−−−−→ M(X4)commutes.40We give a direct proof of the following theorem, which can also be deducedfrom an application of [19, Theorem 5.15] to the category dn.Theorem 3.1.7. Let R : dn→ Ab be a pre-Mackey functor then lims←R∗ =0 for s > 0.Proof. We do induction on n. The case n = 0 is trivial since the complex(3.2) is concentrated at degree 0. There are inclusions of categoriesι0 : dn− 1→ dndefined by [i1 < ... < ik] 7→ [i1 + 1 < ... < ik + 1] andι1 : dn− 1→ dnwhere [i1 < ... < ik] 7→ [0 < i1 +1 < ... < ik+1]. Let C∗dn denote the complexC∗dn(R) in (3.2) that is Ckdn =∏σkR(σk). Define a filtration C∗−1 ⊂ C∗0 ⊂ C∗dnsuch thatCk0 =∏σk=[0<i1<...<ik]R(σk)and Ck−1 = Ck0 for k > 0 and C0−1 = 0. Then by induction Hi(C∗dn/C∗0) =H i(C∗dn−1(R ◦ ι0)) = 0 for i > 0 and Hi(C∗−1) = Hi−1(C∗dn−1(R ◦ ι1)) = 0 fori > 1. And short exact sequences associated to the filtration C∗−1 ⊂ C∗0 ⊂ C∗dnimplies that it suffices to prove H1(C∗0) = 0.For a simplex σ = [0 < i < ... < j] and morphisms αi : σ → [0 < i] andαj : σ → [0 < j] in dn, set φσ = R∗(αj)R∗(αi).For a fixed i, define a map Φi : R([0 < i])→ R([0]) byΦi =∑σ=[0<i<...<j](−1)|σ|−1R∗(dj1)φσ where dj1 : [0 < j]→ [0],where |σ| denotes the dimension of the simplex σ. Combining these maps for41all 1 ≤ i ≤ n define Φ :∏nl=1R([0 < l])→ R([0]) byΦ =n∑i=1Φipii,where pii denotes the natural projection∏nl=1R([0 < l])→ R([0 < i]) to theith factor.We need to show that if U is in the kernel of δ1, then δ0Φ(U) = U orequivalently R∗(di1)Φ(U) = pii(U) for all 1 ≤ i ≤ n. Fix k, and compute thecompositionR∗(dk1)Φ =n∑i=1∑σ=[0<i<...<j](−1)|σ|−1R∗(dk1)R∗(dj1)φσ= pik +n∑i=1∑σ|k∈σ 6=[0<k](−1)|σ|−1R∗(dk1)R∗(dj1)φσ+∑σ|k/∈σ(−1)|σ|−1R∗(dk1)R∗(dj1)φσin the second line first use the identity R∗(dk1)R∗(dj1)φ[0<k] = pik then separatethe summation into two parts such that it runs over the simplices whichcontain {k} and do not contain {k}. Suppose that σ = [0 < i < ... < j] doesnot contain {k} and σ denote the simplex obtained from σ by adjoining {k}.There are three possibilities for σ:σ =[0 < i < ... < k < ... < j] if i < k < j,[0 < i < ... < j < k] if j < k,[0 < k < i < ... < j] if k < i.The corresponding terms of the first two cases cancel out when σ and σ are42paired off in the summation, since the following diagram commutesR(σ) ←−−− R([0 < i])R∗(αi)−−−−→ R(σ)y R∗(αj)yR([0 < k])R∗(dk1)←−−−− R([0])R∗(dj1)←−−−− R([0 < j])and the corresponding terms become equal with opposite signs. So the equa-tion simplifies toR∗(dk1)Φ = pik +∑σ=[0<k<i<...<j](−1)|σ|−1(R∗(dk1)R∗(dj1)φσ −R∗(dk1)R∗(dj1)φd1σ)where the sum runs over the simplices starting with {0 < k}. The mapsσθ→ [0 < k < i], σϑ→ [0 < k] and also the two face maps [0 < i]d1← [0 < k <i]d2→ [0 < k] in our indexing category dn give a commutative diagramR([0 < i])R∗(d1)−−−−→ R([0 < k < i])R∗(θ)−−−→ R(σ)y R∗(ϑ)yR(d1σ) −−−→ R([0 < j])R∗(dk1)R∗(dj1)−−−−−−−−→ R([0 < k]).Hence we can writeR∗(dk1)Φ = pik +∑σ=[0<k<i<...<j](−1)|σ|−1R∗(ϑ)R∗(θ) (R∗(d2)−R∗(d1)) .Now suppose that U is in the kernel of δ1 : C10 → C20 , the summation abovevanishes since (R∗(d2)−R∗(d1)) (U) = 0. Therefore R∗(dk1)Φ(U) = pik(U),which implies that H1(C∗0) = 0.433.1.1 Higher limits of the representation ring functorLet G be a finite group and p be a prime dividing its order. Let M0, ...,Mndenote the maximal abelian p-subgroups of G. Consider the contravariantfunctor R : dn→ Ab defined by σ 7→ R(Mσ) where Mσ = ∩i∈σMi and R(H)is the Grothendieck ring of complex representations of H. Note that Q⊗ Ris a pre-Mackey functor: If α : σ → σ′ set A = ∩i∈σMi and B = ∩i∈σ′Mithen (Q ⊗ R∗(α),Q ⊗ R∗(α)) = (resB,A, 1|B:A| indA,B). Hence Theorem 3.1.7implies the following.Corollary 3.1.8. If R : dn → Ab defined as above then lims←Q ⊗ R∗ = 0for s > 0.Proof. We give an alternative proof which is specific to the representationring functor. Let U : Grp→ Set denote the forgetful functor which sends agroup to the underlying set. The isomorphism C⊗R(Mσ) ∼= HomSet(Mσ,C) =H0(U(Mσ);C)) induces an isomorphism lims←C⊗R(Mσ) ∼= lims←H0(U(Mσ);C).Note that U(Mσ) is a finite set of points. Consider the space hocolimdnU(Mσ)and the associated cohomology spectral sequenceEs,t2 = lims←dnH t(U(Mσ);C)⇒ Hs+t(hocolimdnU(Mσ);C).Now, the diagram dn→ Top defined by σ 7→ U(Mσ) is free (Appendix), con-sists of inclusions of finite sets. Therefore the natural map hocolimdnU(Mσ)→colimdnU(Mσ) = U(G) is a homotopy equivalence, note that the latter is a fi-nite set. The spectral sequence collapses atE2-page and gives lims←H0(U(Mσ);C) ∼=Hs(hocolimdnU(Mσ);C) which vanishes for s > 0.443.2 K-theory of B(2, G)In this section, we assume that G is a finite group. We study the complexK-theory of B(q,G) when q = 2. Theorem 1.2.6 gives a decompositionK˜∗(B(2, G)) ∼=⊕p||G|K˜∗(B(2, G)p),of the reduced K-theory of B(2, G). For each such prime p recall the fi-bration hocolimN (q,G)pG/P → B(q,G)p → BG from §1.2. Furthermore, the Borelconstruction commutes with homotopy colimits and gives a weak equivalenceB(q,G)p ' (hocolimN (q,G)pG/P )×G EG.There is the Atiyah-Segal completion theorem [4] which relates the equivari-ant K-theory of a G-space X to the complex K-theory of its Borel construc-tion X ×G EG.Equivariant K-theoryComplex equivariant K-theory is a Z/2-graded cohomology theory definedfor compact spaces using G-equivariant complex vector bundles [24]. RecallthatKnG(G/H) ∼={R(H) if n = 0,0 if n = 1,where H ⊆ G and recall that R(H) denotes the Grothendieck ring of complexrepresentations of H. Note that in particular, K0G(pt) ∼= R(G) and K∗G(X)is an R(G)-module via the natural map X → pt.Let X be a compact G-space and I(G) denote the kernel of the augmen-tation map R(G)ε→ Z. The Atiyah-Segal completion theorem [4] states thatK∗(X ×G EG) is the completion of K∗G(X) at the augmentation ideal I(G).45In particular, taking X to be a point this theorem implies thatKn(BG) ∼={R(G)∧ n = 0,0 n = 1.The completion R(G)∧ can be described using the restriction maps to a Sylowp-subgroup for each p dividing the order of the group G. Let Ip(G) denotethe quotient of I(G) by the kernel of the restriction map I(G)→ I(G(p)) toa Sylow p-subgroup. There are isomorphisms of abelian groupsKn(BG) ∼={Z⊕⊕p| |G| Zp ⊗ Ip(G) n = 0,0 n = 1,see [21]. Note that if G is a nilpotent group, equivalently G ∼=∏p| |G|G(p),the restriction map I(G)→ I(G(p)) is surjective andK˜0(BG) ∼= I(G)∧ ∼=⊕p| |G|Zp ⊗ I(G(p)). (3.3)We also use the fact that the completion of R(H) with respect to I(H) isisomorphic to the completion with respect to I(G) as an R(G)-module viathe restriction map R(G)→ R(H).3.2.1 K-theory of B(2, G)pThe E2-page of the equivariant version of the Bousfield-Kan spectral sequence[20] for the equivariant K-theory of hocolimN (q,G)pG/P is given byEs,t2 ∼= lims←N (q,G)pKtG(G/P ) ∼=lims←N (q,G)pR(P ) if t is even,0 if t is odd,(3.4)46and the spectral sequence converges to K∗G(hocolimN (q,G)pG/P ) whose completionis K∗(hocolimN (q,G)pBP ). We will split off the trivial part of Es,t2 and considerE˜s,t2 ∼= lims←N (q,G)pI(P ) for t even.Note that the splitting R(P ) ∼= Z ⊕ I(P ) gives lims←N (q,G)pR(P ) ∼= lims←N (q,G)pI(P )for s > 0 since the nerve of N (q,G)p is contractible. The following resultdescribes the terms of this spectral sequence for the case q = 2.Lemma 3.2.1. Let R : N (2, G)p → Ab be the representation ring functordefined by P 7→ R(P ) then lims←R are torsion groups for s > 0. Furthermore,lim0←R ∼= Znp+1 where np is the number of (non-identity) elements of ordera power of p in G.Proof. Let Mi for 0 ≤ i ≤ n denote the maximal abelian p-subgroups of G.Define a functor g : dn→ N (2, G)p by sending a simplex [i0 < i1 < ... < ik]to the intersection⋂kj=0Mij . Then g has a left adjoint, namely the functorP 7→⋂Mi:P⊂MiMi sending an abelian p-subgroup to the intersection of max-imal abelian p-subgroups containing it. By Proposition 3.1.3 we can replacethe indexing category N (2, G)p by dn and calculate the higher limits overthe category dn. Then the first part of the theorem follows from Corollary3.1.8.For the second part note that lim0←R = lim←R is a submodule of a freeZ-module, hence it is free. Then it suffices to calculate its rank. Tensoringwith C gives an isomorphismC⊗ lim0←N (2,G)pR(P ) ∼= HomSet(∪N (2,G)pP,C).Recall that N (2, G)p is the collection of abelian p-subgroups of G. The result47follows from the identity | ∪N (2,G)p P | = np + 1 where np is the number of(non-identity) elements of order a power of p in G.Theorem 3.2.2. There is an isomorphismQ⊗Ki(B(2, G)) ∼={Q⊕⊕p||G|Qnpp if i = 0,0 if i = 1,where np is the number of (non-identity) elements of order a power of p inG.Proof. By the decomposition in Theorem 1.2.6 we work at a fixed prime pdividing the order of G. Then K∗(B(2, G)p) is isomorphic to the completionof K∗G(hocolimN (2,G)pG/P ) at the augmentation ideal I(G) by the Atiyah-Segalcompletion theorem. Since R(G) is a Noetherian ring the completion −⊗R(G)R(G)∧ is exact on finitely generated R(G)-modules, hence commutes withtaking homology. Moreover, (3.3) givesR(P )⊗R(G) R(G)∧ ∼= R(P )∧ ∼= Z⊕ I(P )⊗ Zp.This isomorphism inducesE˜s,tr ⊗I(G) I(G)∧ ∼= E˜s,tr ⊗Z Zp for r ≥ 2,which gives an isomorphism between the abutments. For t even and s > 0,E˜s,t2 are torsion by Lemma 3.2.1. After tensoring with Q, the spectral se-quence collapses onto the vertical axis at the E2-page andQ⊗K˜0(B(2, G)p) ∼=Q⊗ Znpp whereas Q⊗K1(B(2, G)p) vanishes.48Chapter 4Examples4.1 Homotopy type of B(2, G)Extraspecial p–groups Srp are key examples we considered in the previoussections, and they turn out to be useful in determining torsion in the kernelof ψ : G(2) → G for other classes of groups. We start with a discussion ofsome properties of B(2, G) for the extraspecial p–groups of rank 4, and thenwe discuss torsion in T (2) for symmetric groups and general linear groups.We want to determine the universal cover B˜(2, S2p) of B(2, S2p), which ishomotopy equivalent to the universal cover Y of E(2, S2p). Recall that thereis an extension Z/pS2p → V = (Z/p)4. As we know from the description ofthe poset I(b) of isotropic subspaces of V the maximal ones has dimension2. Cohomology groups of the universal cover Y can be calculated from thechain complex∏W0←W1←W2Z[V (b)/W2]→∏W0←W1Z[V (b)/W1]→∏W0Z[V (b)/W0] (4.1)where Wi ∈ I(b) are of dimension 0 ≤ i ≤ 2. The space Y is a simplyconnected 2–dimensional complex, in particular its cohomology is torsion49free. Therefore there is an isomorphismpii(B(2, S2p)) ∼= pii(d∨S2) for i > 1 for some d ≥ 0.The dimension d can be computed from the Euler characteristic, for examplein the case of p = 2 we haveχ(B˜(2, S22)) =642|{0 ⊂ W1 ⊂ W2}| −642|W1||{W1 ⊂ W2}|+642|W2||{W2}|=64245−(64445 +64230)+(64815 +64415 +642)= 152,therefore d = 151. In 4.1 the group V (b) ∼= (Z/2)5 has order 64/2 (§2.3.1),and the number of flags can be counted inductively. See [26] for a countingof the number of flags in the symplectic space 2H = ((Z/2)4, b).Next we give some class of groups which satisfy the conclusion of Theorem2.4.5, and therefore contain torsion in the kernel of the natural map ψ :G(2) → G. We refer the reader to the relevant sections in [2] for the grouptheoretic facts used below.1. The general linear group GLn(Fq), n ≥ 4, over the finite field Fq ofcharacteristic p has a symplectic sequence given by the elementary ma-trices{g1 = E12, g2 = E13, g3 = E2n, g4 = E3n}where Eij has 1’s on the diagonal and in the (i, j)-slot. It follows eas-ily from the commutation relations of elementary matrices that thissequnce satisfies the required commutation relations of a symplecticsequence. The commutator E1n = [g1, g3] has order q, hence by Theo-rem 2.4.5 the kernel of GLn(Fq)(2) → GLn(Fq) has a torsion elementof order p.502. Next we consider the n–fold wreath product Wp(n) = Z/p o · · · o Z/p︸ ︷︷ ︸nfor n ≥ 3. Regarding linear transformations of an n dimensional vectorspace V over Fp as the permutations of a set of cardinality pn gives aninclusionι : GLn(p)→ Σpn .Consider the case of n = 4, and the images of the elements in {gi}4i=1under this embedding. The elements in V fixed by all gi for 1 ≤ i ≤ 4is given by the subspace {(x, 0, 0, 0)| x ∈ Fp} whose cardinality is p.Hence the images land inside the symmetric group Σp4−p ⊂ Σp4 underthe inclusion ι. The p–Sylow subgroup S =∏p−1(Wp(3) × Wp(2) ×Wp(1)) of Σp4−p contains the image of the symplectic sequence. There-fore there is a p torsion in the kernel of ψS : S(2)→ S. Note that theisomorphism (G×H)(2) ∼= G(2)×H(2) for the product of any given twogroups G and H induces an isomorphism kerψG×H ∼= kerψG × kerψH .Since S is a product of wreath products, the kernel kerψWp(3) whichcorresponds to the largest wreath product contains a p torsion. Thistorsion element comes from the symplectic sequence {pi(gi)}4i=1 wherepi : S → Wp(3) is the natural projection.3. Sylow p–subgroups of the symmetric group Σn on n letters is describedby the p–adic expansion of n. For example Sylp(Σpk) is the k–foldwreath product okZ/p. The previous example implies that Σn withn =∑mi=1 αipi where 0 ≤ αi ≤ p − 1 contains a p torsion in kerψΣnwhen m ≥ 3. This is a consequence of the inclusion omZ/p ⊂ Σn.One can say the same thing for the alternating group An. For an oddprime p both have the same Sylow subgroups Sylp(An) = Sylp(Σn), andthe case p = 2 can be settled by the isomorphism GL4(F2) ∼= A8.514.2 K–theoryWe consider the class of groups called TC–groups, these groups are studiedin [1]. A group G is called a transitively commutative group (TC-group) ifcommutation is a transitive relation for non-central elements that is [x, y] =1 = [y, z] implies [x, z] = 1 for all x, y, z ∈ G − Z(G). Any pair of maximalabelian subgroups of G intersect at the center Z(G).We want to discuss the complex K–theory of B(2, G) for this class ofgroups. Let M1, · · · ,Mk denote the set of maximal abelian subgroups of G.Higher limits of the representation ring functor R : N (2, G) → Ab fit intoan exact sequence0→ lim0←R→ R(Z(G))⊕k⊕i=1R(Mi)θ→k⊕i=1R(Z(G))→ lim1←R→ 0where θ(z,m1, ...,mk) = (z − resM1,Z(G)m1, ..., z − resMk,Z(G)mk). The mapθ is surjective since the restriction maps R(Mi) → R(Z(G)) are surjective.Therefore no higher limit occurs, i.e. lim1←R = 0. The same conclusion holdsfor the restriction of the functor R to N (2, G)p ⊂ N (2, G). Therefore thespectral sequence in (3.4) collapses at the E2-page and after completionKi(B(2, G)) ∼={Z⊕⊕p||G| Znpp if i = 0,0 if i = 1.For extraspecial p–groups of rank 2r ≥ 4, non-vanishing higher limits ofthe representation ring functor occur in the Bousfield-Kan spectral sequence.In the case of S22 a calculation in GAP [15] shows thatlimi←N (2,S22)R ∼=Z32 i = 0,(Z/2)9 i = 1,0 otherwise.52Therefore the spectral sequence in (3.4) collapses at the E2-page and aftercompletionKi(B(2, S22)) ∼={Z⊕ Z312 if i = 0,(Z/2)9 if i = 1.At the prime p = 3 on the other hand one hasKi(B(2, S23)) ∼={Z⊕ Z2423 if i = 0,(Z/3)177 if i = 1.Note that the K-theory computation also confirms that B(S2p , 2), for p = 2and 3, is not homotopy equivalent to B(Ŝ2p), as the latter has K1(B(Ŝ2p)) = 0as a consequence of the Atiyah-Segal completion theorem.53Bibliography[1] A. Adem, F.R. Cohen, E. Torres-Giese: Commuting elements,simplicial spaces and filtrations of classifying spaces, Math. Proc. Cam.Phil. Soc. Vol. 152 (2011), 91-114.[2] A. Adem, R. J. Milgram: Cohomology of finite groups, Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathe-matical Sciences], 309. Springer-Verlag, Berlin, 2004.[3] M. Aschbacher: Finite group theory, Cambridge Studies in AdvancedMathematics, 10. Cambridge University Press, Cambridge, 1986.[4] M. F. Atiyah, G.B. Segal: Equivariant K-Theory and Completion,J. Differential Geom. Volume 3, Number 1-2 (1969), 1-18.[5] D.J. Benson, S.D. Smith: Classifying spaces of sporadic groups, Math-ematical Surveys and Monographs, 147. American Mathematical Society,Providence, RI, 2008.[6] A.K. Bousfield, D.M. Kan: Homotopy Limits, Completions and Lo-calizations, Lecture Notes in Mathematics, Vol. 304. Springer-Verlag,Berlin-New York, 1972.[7] K. Brown: The coset poset and probabilistic zeta function of a finitegroup, J. Algebra 225 (2000), no. 2, 989-1012.54[8] K. Brown: Cohomology of groups, Graduate Texts in Mathematics, 87.Springer-Verlag, New York-Berlin, 1982.[9] J.D. Dixon, M.P.F. Du Sautoy, A. Mann, D. Segal: AnalyticPro-p Groups, Cambridge Studies in Advanced Mathematics, 61. Cam-bridge University Press, Cambridge, 1999.[10] A. W. M. Dress: Contributions to the theory of induced represen-tations, Algebraic K-theory, II: ”Classical” algebraic K-theory and con-nections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle,Wash., 1972), pp. 183-240. Lecture Notes in Math., Vol. 342, Springer,Berlin, 1973.[11] W. G. Dwyer: Classifying spaces and homology decompositions,Topology 36 (1997), no. 4, 783-804.[12] R. Elman, N. Karpenko, A. Merkurjev: The algebraic and ge-ometric theory of quadratic forms, American Mathematical Society Col-loquium Publications, 56. American Mathematical Society, Providence,RI, 2008.[13] E. D. Farjoun: Fundamental group of homotopy colimits, Adv. Math.182 (2004), no. 1, 127.[14] E. D. Farjoun: Cellular spaces, Nullspaces and Homotopy localization,Springer LNM 1622 (1996)[15] The GAP Group: GAP – Groups, Algorithms, and Programming,Version 4.4.12, 2008, ([16] P. G. Goerss, J. F. Jardine: Simplicial homotopy theory, Progressin Mathematics, 174. Birkhuser Verlag, Basel, 1999.[17] J. Grodal: Higher limits via subgroup complexes, Ann. of Math. (2)155 (2002), no. 2, 405-457.55[18] P.S. Hirschhorn: Model categories and their localizations, Mathe-matical Surveys and Monographs, 99. American Mathematical Society,Providence, RI, 2003.[19] S. Jackowski, J. McClure: Homotopy decomposition of classify-ing spaces via elementary abelian subgroups., Topology 31 (1992), no. 1,113132.[20] C. N. Lee: A homotopy decomposition for the classifying space of virtu-ally torsion-free groups and applications., Math. Proc. Cambridge Philos.Soc. 120 (1996), no. 4, 663-686.[21] W. Lu¨ck: Rational computations of the topological K-theory of classify-ing spaces of discrete groups (English summary), J. Reine Angew. Math.611 (2007), 163-187.[22] N. Nikolov, D. Segal: Generators and commutators in finite groups;abstract quotients of compact groups. (English summary), Invent. Math.190 (2012), no. 3, 513-602.[23] B. Oliver: Higher Limits Via Steinberg Representations, Comm. Al-gebra 22 (1994), no. 4, 138-1393.[24] G. Segal: Equivariant K-Theory, Inst. Hautes tudes Sci. Publ. Math.No. 34 1968 129-151.[25] J. P. Serre: Trees, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.[26] S. D. Smith: Subgroup complexes Mathematical Surveys and Mono-graphs, 179. American Mathematical Society, Providence, RI, 2011.[27] P. Webb: An introduction to the representations and cohomology ofcategories, Group representation theory, 149-173, EPFL Press, Lausanne,2007.56[28] V. Welker, G. Ziegler, R. T. Zˇivaljevic´: Homotopy colimits—comparison lemmas for combinatorial applications, J. Reine Angew.Math. 509 (1999), 117-149.57AppendixWe review some definitions about simplicial sets especially homotopy colim-its. We refer the reader to [16] and [11].The category of simplicial sets S and the category of topological spacesTop are closely related. Given T a topological space there is a functorTop→ S called the singular set defined bySing(T ) : n 7→ HomTop(|∆n|, T )where |∆n| is the topological standard n-simplex. Conversely a simplicial setX gives rise to a topological space via the geometric realization functor givenby the difference cokernel|X| = lim→{∐θ:m→nXn × |∆m|⇒∐nXn × |∆n|}where (x, p) 7→ (x, θ∗(p)) and (x, p) 7→ (θ∗(x), p), see [28]. There is an adjointrelation between these two functors|.| : S  Top : Sing.Note that |.| preserves all colimits in S since it has a right adjoint.Let I be a small category, and i ∈ I an object, the under category I \ i isthe category whose objects are the maps i→ i0 and morphisms from i→ i0to i→ i1 are commutative diagramsi > i0i1.∨>Let F : I→ Top be a functor. The homotopy colimit of F is the topological58space described as a difference cokernelhocolim F = lim→{∐f :j→iB(I \ i)× F (j) ⇒∐iB(I \ i)× F (i)}where (b, y) 7→ (B(f)(b), y) and (b, y) 7→ (b, f(y)), see [28]. If one wants towork in the category of simplicial sets, given F : I → S one can use thesimplicial set B•(I \ i) instead.From the definition we see that there is a natural map to the ordinarycolimithocolim F → colim Fobtained by collapsing the nerves B(I \ i) → pt. This map is not usuallya weak equivalence. In general, it is a weak equivalence if the diagram F :I → S is a free diagram ([14, Appendix HC]). A diagram of sets I → Setis free if it is of the form∐i Fi for a collection of subobjects of I whereeach diagram F i : I → Set is defined by F i(j) = HomI(i, j). A diagram ofsimplicial sets is free if in each dimension it gives a free diagram of sets. Moreprecisely, a diagram F : I→ S is free if there exists a sequence {S0, S1, · · ·}of diagrams Sn : Iδ → Set such that Sn(i) ⊂ F (i)n, si(Sn(i)) ⊂ Sn+1(i)for each degeneracy map si, and for σ ∈ F (i)n there exists a morphismγ : j → i in I and τ ∈ Sn(j) such that F (γ)(τ) = σ ([18, 14.8.4]). Themorphism γ and the n–simplex τ are supposed to be unique. Here Iδ denotesthe discrete category which consists of the same objects as I and only theidentity as morphisms. In particular, a poset of simplicial sets which isclosed under taking subsets and partially ordered by inclusion gives a freediagram of simplicial sets. Let F : I→ S be such a diagram. In this case setSn(i) = F (i)n−∪γ:jiF (γ)(F (j)n). Then given σ ∈ F (i)n choose j such thatF (j) = ∩σ∈F (k)nF (k), such a j exists since we assume the diagram is closedunder intersections. The unique morphism γ : j → i gives F (γ)(τ) = τ .Throughout the thesis we use diagrams indexed by a collection of groups59closed under intersections, and after applying the classifying space functorthis gives a diagram of simplicial sets closed under intersection. Hence thosekind of diagrams are free.A standard example of a homotopy colimit is the Borel construction. LetX be a G-space and consider G the category associated to the group thenhocolim F ' X ×G EGwhere F : G → Top sends the single object to X. The most importantproperty of homotopy colimits is that a natural transformation F → F ′ offunctors such that F (i)→ F ′(i) is a weak equivalence for all i ∈ I induces aweak equivalencehocolim F → hocolim F ′.Another useful property we use in the thesis is the commutativity of homo-topy colimits. We refer the reader to [28] for further properties of homotopycolimits.60


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