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A simplicial approach to spaces of homomorphisms Villarreal Herrera, Bernardo 2017

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A Simplicial Approach to Spaces of HomomorphismsbyBernardo Villarreal HerreraBSc Mathematics, Universidad Nacional Auto´noma de Me´xico, 2011MSc Mathematics, Universidad Nacional Auto´noma de Me´xico, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mathematics)The University of British Columbia(Vancouver)August 2017c© Bernardo Villarreal Herrera, 2017AbstractLet G be a real linear algebraic group and L a finitely generated cosimplicial group.We prove that the space of homomorphisms Hom(Ln,G) has a homotopy stabledecomposition for each n ≥ 1. When G is a compact Lie group, we show thatthe decomposition is G-equivariant with respect to the induced action of conju-gation by elements of G. In particular, under these hypotheses on G, we obtainstable decompositions for Hom(Fn/Γqn,G) and Rep(Fn/Γqn,G) respectively, whereFn/Γqn are the finitely generated free nilpotent groups of nilpotency class q−1. Thespaces Hom(Ln,G) assemble into a simplicial space Hom(L,G). When G =U weshow that its geometric realization B(L,U) has a non-unital E∞-ring space structurewhenever Hom(L0,U(m)) is path connected for all m≥ 1.We describe the connected components of Hom(Fn/Γqn,SU(2)) arising fromnon-commuting q-nilpotent n-tuples. We prove this by showing that all these n-tuples are conjugated to n-tuples consisting of elements in the the generalizedquaternion groups Q2q ⊂ SU(2), of order 2q. Using this result, we exhibit thehomotopy type of ΣHom(Fn/Γqn,SU(2)) and a homotopy description of the clas-sifying spaces B(q,SU(2)) of transitionally q-nilpotent principal SU(2)-bundles.The above computations are also done for SO(3) and U(2).Finally, for q = 2, the space B(2,G) is denoted BcomG, and we compute theintegral cohomology ring for the Lie groups G= SU(2) and U(2). We also includecohomology calculations for the spaces BcomQ2q .iiLay SummaryIn mathematics, Algebraic Topology focuses in developing algebraic tools to un-derstand geometric objects such as shapes and graphs, or in general, abstract spaces.In this thesis we study these objects with a given binary operation, by analyzing theelements of the object that are symmetric with respect to the operation. One maycertainly has encountered with such symmetries in binary operations during theirearly studies of mathematics, specifically with the Commutative Law of additionand multiplication of the natural numbers. This is a global property of the natu-ral numbers that makes them easy to handle. To describe broader objects we usemore complex operations that fail to be globally commutative (for example matrixmultiplication). Although, we can consider the “space” of commutative pairs andextract very interesting information as we show in this work.iiiPrefaceThe results in Chapter 1 constitute the research done during the first years of mygraduate program, under the supervision of Prof. A. Adem, and has been acceptedfor publication in [28].The results in Chapter 2 and the last part of Chapter 3 is joint work with O.Antolı´n Camarena in [8], and has been submitted for publication.The first part of Chapter 3 is work in progress with O. Antolı´n Camarena andSimon Gritschacher.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Spaces of Homomorphisms via Cosimplicial groups . . . . . . . . . 41.1 Homotopy Stable Decompositions . . . . . . . . . . . . . . . . . 71.1.1 Spaces of Homomorphisms . . . . . . . . . . . . . . . . 71.1.2 Triangulation of Semi-algebraic Sets . . . . . . . . . . . . 91.1.3 Simplicial Spaces and Homotopy Stable Decompositions . 101.1.4 Cosimplicial Groups, 1-cocycles and Hom(L,G) . . . . . 111.1.5 Homotopy Stable Decomposition of Hom(Ln,G) . . . . . 171.1.6 Equivariant Homotopy Stable Decomposition of Hom(Ln,G) 211.2 Homotopy Properties of B(L,G) . . . . . . . . . . . . . . . . . . 251.2.1 Geometric realization of Hom(L,G) . . . . . . . . . . . . 251.2.2 Relation between commutative I-monoids and infinite loopspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2.3 Non-unital E∞-ring space structure of B(L,U) . . . . . . . 31v2 Nilpotent n-tuples in SU(2) . . . . . . . . . . . . . . . . . . . . . . . 352.1 Non-abelian nilpotent subgroups of SU(2) . . . . . . . . . . . . . 372.1.1 Consequences for SO(3) and U(2) . . . . . . . . . . . . . 432.1.2 2-nilpotent tuples in U(m) . . . . . . . . . . . . . . . . . 472.2 Stable Homotopy type . . . . . . . . . . . . . . . . . . . . . . . 492.3 Homotopy type of B(q,SU(2)) . . . . . . . . . . . . . . . . . . . 543 Cohomology Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1 The space BcomG1 . . . . . . . . . . . . . . . . . . . . . . . . . . 583.1.1 Integral Cohomology ring of BcomSO(3)1 . . . . . . . . . 593.1.2 Integral Cohomology ring of BcomU(2) and BcomSU(2) . . 633.2 F2-Cohomology ring of BcomQ2q . . . . . . . . . . . . . . . . . . 68Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A.1 Computing ker f in SINGULAR . . . . . . . . . . . . . . . . . . . 73viAcknowledgementsI would specially like to thank A. Adem for his supervision and expert advicethroughout this work. O. Antolı´n Camarena for his useful input within the detailsof the results of Chapter 1 and for a fruitful collaboration for the past 2 years.I would also like to thank K. Karu for pointing out the necessity of Lemma1.1.18. B. Williams for making me notice a crucial fact about spectral sequencesand its impact in the results of Chapter 3. Jose´ Manuel Go´mez for helpful sug-gestions about U(2) and for motivating me and O. Antolı´n to look into SU(m)and U(m) for m > 2. Simon Gritschacher for verifying part of the calculations inChapter 3.A special mention to M. Bergeron, F. Cohen and J. M. Go´mez for their com-ments on an earlier version of Chapter 1.Finally, to my financial sponsor CONACyT.viiIntroductionIn the realm of algebraic topology, one of the roles of a Lie group arises whenstudying locally trivial fibrations through their structure group. For example, realvector bundles of rank n over a space X have a subgroup G of the general lineargroup GLn(R) as structure group. A classical result in homotopy theory is thatisomorphism classes of such vector bundles are in bijection with homotopy classesof maps [X ,BG], where the space BG is the classifying space. When X is a smoothmanifold, we can endow any vector bundle E of rank n over X with a smoothstructure. Giving a “flat connection” on E is equivalent to factoring the associatedmap X → BG, through X → BGδ → BG, where Gδ is G seen as a discrete group.Moreover, if X is aspherical and connected (e.g. an oriented surface of genus graterthan 1, the torus of rank k), then [X ,BGδ ] is in bijection with the set of conjugationclasses of group homomorphisms of the fundamental group of X into G, denotedHom(pi1(X),G)/G. In other words, the set of homomorphisms Hom(pi1(X),G)carries geometrical information of vector bundles over X , despite only using thegroup structure of G. Thus, it is natural to try to understand how the topology of Ginterplays with group homomorphisms Γ→ G, where Γ is a discrete group.We can topologize Hom(Γ,G), where Γ is a finitely generated discrete group,by choosing a set of generators on Γ, {a1, ...,an}, and then identifying Hom(Γ,G)with the subspace of Gn that consists of the ordered n-tuples (ρ(a1), ...,ρ(an))where ρ : Γ→ G is a homomorphism. Once we have a topology on this set, wemay ask if Hom(Γ,G) shares the same topological properties as G. For example,if G has a CW-complex structure, does Hom(Γ,G)? We fully address this questionin Chapter 1, and it turns out that when G is a real linear algebraic group, thenHom(Γ,G) is an affine variety, and thus can be triangulated. In fact, we study the1following situation. Let X be a CW-complex and A ⊂ X a subcomplex. When-ever there is a retraction A ↪→ X → A, the inclusion splits after one suspension,that is, ΣX is homotopy equivalent to the wedge of spaces ΣX/A∨ΣA. We explainunder what circumstances Hom(Γ,G) admits stable splittings. This problem hasbeen studied before in several cases. When G ⊂ GLn(C) is a closed subgroup, A.Adem and F. Cohen (in [2]) gave a homotopy stable decomposition for Hom(Zn,G)as wedges of the quotient spaces Hom(Zk,G)/S1(Zk,G) with 1 ≤ k ≤ n. HereS1(Zk,G) stands for the k-tuples with at least one entry equal to the identity ma-trix I in G. For an arbitrary finitely generated abelian group pi , A. Adem andJ. M. Go´mez (in [6]) gave a similar stable decomposition for Hom(pi,G), but inthis case, G is a finite product of the compact Lie groups SU(r),Sp(k) and U(m).We show that the homotopy stable decomposition in [2] (in general, a version ofit) still holds if we replace the family {Zn}n≥1 with a broader object, namely, afinitely generated cosimplicial group L, that is, for every n ≥ 0, we have a finitelygenerated group Ln with its coface and codegeneracy homomorphisms. A specificexample of L, comes from the finitely generated free nilpotent groups Fn/Γqn. HereFn denotes the free group on n-generators and Γqn is the q-th stage of its descend-ing central series. We work under the assumption that G is a real linear algebraicgroup, and with it we can prove that the inclusions Sk+1(Fn/Γqn,G) ↪→ Sk(Fn/Γqn,G)are closed cofibrations for every 0 ≤ k ≤ n, where Sk(Fn/Γqn,G) denotes the sub-space of Hom(Fn/Γqn,G) with at least k entries equal to the identity element in G.This condition as stated by A. Adem, F. Cohen and E. Torres-Giese in [4, p. 102]says that real linear algebraic groups have cofibrantly filtered elements, and theyshow that this implies Hom(Fn/Γqn,G) splits after one suspension as wedges ofHom(Fk/Γqk ,G)/S1(Fk/Γqk ,G) where 0 ≤ k ≤ n. This decomposition was knownbefore for some compact and connected Lie groups G (see for example F. Cohenand M. Stafa in [13, Remark 1, p. 387, Theorem 2.13, p. 388]).In the second chapter we address a more classical question concerning path-connectedness. It is plausible to think that for compact and connected Lie groupsG, Hom(Zn,G) is also connected. This is true for the Lie groups SU(n), U(m)and Sp(k). But interestingly enough, this is not the case in general, for exampleHom(Zn,SO(3)) is not connected. This is due to the fact that the dihedral subgroupD4 is not contained in a connected abelian subgroup of SO(3). The inverse image2of D4 under the double covering map SU(2)→ SO(3) is the quaternion group Q8,which is no longer abelian, but 2-nilpotent. Therefore it is not surprising that thespace of 2-nilpotent tuples Hom(Fn/Γ3n,SU(2)) is no longer connected. Notice thatthe descending central sequence induces a filtrationHom(Zn,G)⊂ Hom(Fn/Γ3n,G)⊂ ·· · ⊂ Hom(Fn/Γqn,G)⊂ ·· · .What is surprising is that in this filtration, as q increases, also do the number ofconnected components. We show this by giving an explicit description of the con-nected components (all turn out to be homeomorphic to RP3), by proving that theq-nilpotent subgroups of SU(2) are all conjugate to the generalized quaternion oforder Q2q+1 . Using the homotopy stable decomposition obtained in Chapter 1, weshow the homotopy type of ΣHom(Fn/Γqn,G) for G = SO(3),SU(2) and U(2).Another interesting feature of spaces of commuting tuples (also shared by thespaces of q-nilpotent tuples) is that the inclusions in : Hom(Zn,G)→ Gn are com-patible with the degeneracy and face maps arising form NG, the nerve of G asa category with one object. In other words, the collection {Hom(Zn,G)}n≥0 as-sembles into a simplicial subspace Hom(Z•,G) of NG, the latter being a simpli-cial model of the classifying space BG already mentioned above. In [5], the au-thors define BcomG := |Hom(Z•,G)|, the geometric realization of Hom(Z•,G). Anintriguing fact about BcomSU(2) is that it has an infinite number of cohomologygroups with 2-torsion, as opposed to the cohomology groups of BSU(2), for whichH∗(BSU(2);Z)∼= Z[c2], where c2 has degree 4. In chapter 3 we analyze this situa-tion by computing the integral cohomology ring. Surprisingly, this 2-torsion arisesfrom the fact that BcomSU(2) has vanishing first Chern class. We finish this chapterby computing the F2-cohomology ring of BcomQ2q using the characterization of thenilpotent subgroups of Q2q described in Chapter 2.3Chapter 1Spaces of Homomorphisms viaCosimplicial groupsThe main propose of this chapter is to give a general treatment of spaces of ho-momorphisms from a cosimplicial point of view, that will allow us to prove ourstable splitting theorem. Instead of using a finitely generated group as a source,we take cosimplicial groups. To be more precise, let ∆ stand for the categorywhose objects are the finite sets {0,1, ...,n} for n ≥ 0 and morphisms are orderpreserving maps. If L : ∆→ Grp is a finitely generated cosimplicial group andG is a topological group, we get the simplicial space Hom(L,G) : ∆op → Top,where Hom(L,G)n := Hom(Ln,G). We give a homotopy stable decompositionfor the n-simplices of Hom(L,G) as follows. Let X be a simplicial space. DefineSt(Xn) as the subspace of Xn in which any element is in the image of the compo-sition of at least t degeneracy maps. We say X is simplicially NDR when all pairs(St−1(Xn),St(Xn)) are neighbourhood deformation retracts. It was proven by A.Adem, A. Bahri, M. Bendersky, F. Cohen, and S. Gitler in [1] that when X is simpli-cially NDR, each Xn is homotopy stable equivalent to wedges of St(Xn)/St+1(Xn)with 0 ≤ t ≤ n. When G is a real linear algebraic group, the simplicial spaceX = Hom(L,G) is simplicially NDR. We prove this by showing that the subspacesSt(Xn) are real affine subvarieties of Xn for all 0≤ t ≤ n and therefore can be simul-taneously triangulated. Denote St(Ln,G) := St(Hom(Ln,G)) and ΣB as the reducedsuspension of a topological space B.4Theorem 1.0.1. Let G be a real linear algebraic group, and L a finitely generatedcosimplicial group. For each n, there are natural homotopy equivalencesΘ(n) : ΣHom(Ln,G)'∨0≤k≤nΣ(Sk(Ln,G))/Sk+1(Ln,G)).The free groups Fn, assemble into a cosimplicial group which we denote by F .In this case Hom(F,G) is NG, the nerve of G seen as a topological category withone object, which is also the underlying simplicial space of a model of the classi-fying space BG. For each n, we take quotients Fn/Kn by normal subgroups Kn thatare compatible with coface and codegeneracy homomorphisms of F to get finitelygenerated cosimplicial groups denoted by F/K. The induced simplicial spaces aremore easily described since there is a simplicial inclusion Hom(F/K,G) ⊂ NG.Fixing q> 0, the family of subgroups Γqn arising from the descending central seriesof Fn, is compatible with F , and induce the finitely generated cosimplicial groupF/Γq. The authors of [4] conjectured that the closed subgroups of GLn(C) havecofibrantly filtered elements and thus the homotopy stable decomposition holds.Applying Theorem 1.0.1 to F/Γq allows us to prove the following version of theconjecture.Corollary 1.0.2. If G is a Zariski closed subgroup of GLn(C), then there arehomotopy equivalences for the cosimplicial group F/Γq,ΣHom(Fn/Γqn,G)'∨1≤k≤nΣ(nk)∨Hom(Fk/Γqk ,G)/S1(Fk/Γqk ,G)for all n and q.For any finitely generated cosimplicial group L, conjugation under elements ofG gives Hom(Ln,G) a G-space structure. Moreover, if G is a real algebraic lineargroup, then it has a G-variety structure. The subspaces St(Hom(Ln,G)) are subva-rieties that are invariant under the action of G for all 0 ≤ t ≤ n. Using techniquesfrom D.H. Park and D.Y. Suh in [26], when G is a compact Lie group, we showthat Hom(Ln,G) has a G-CW-complex structure, where each St(Hom(Ln,G)) isa G-subcomplex. This allows us to prove the equivariant version of the previous5theorem. Let Rep(Ln,G) and St(Ln,G) denote the orbit spaces of Hom(Ln,G) andSt(Hom(Ln,G)) respectively.Theorem 1.0.3. Let G be a compact Lie group. Then for each n, Θ(n) in Theorem1.0.1 is a G-equivariant homotopy equivalence, and in particular we get homotopyequivalencesΣRep(Ln,G)'∨1≤k≤nΣ(Sk(Ln,G)/Sk+1(Ln,G)).Applying this to the cosimplicial group F/Γq as in Corollary 1.0.2, we obtainΣRep(Fn/Γqn,G)'∨1≤k≤nΣ(nk)∨Rep(Fk/Γqk ,G)/S1(Fk/Γqk ,G) .In the second part of this chapter we study the geometric realization of the sim-plicial space Hom(L,G) for a finitely generated cosimplicial group L, and denoteit by B(L,G). We show that the set of 1-cocycles of L, Z1(L), is in one to onecorrespondence with cosimplicial morphisms F → L. With this we show that any1-cocycle of L defines a principal G-bundle over B(L,G) which is the pullback ofthe universal bundle EG→ BG by the induced map B(L,G)→ BG.When G=U = colimmU(m)we show that B(L,U) has an I-rig structure, that is,if I stand for the category of finite sets and injections, the functor B(L,U( )) : I→Top is symmetric monoidal with respect to both symmetric monoidal structures onI. Using the machinery developed in [7] by A. Adem, J. M. Go´mez, J. Lind and U.Tillman, we prove:Theorem 1.0.4. Let L be a finitely generated cosimplicial group and suppose thatthe space Hom(L0,U(m)) is path connected for all m ≥ 1. Then, B(L,U) is anon-unital E∞-ring space.This theorem is also true if we replace U by SU,Sp,SO or O.61.1 Homotopy Stable Decompositions1.1.1 Spaces of HomomorphismsLet G be a topological group and Γ a finitely generated group. Any homomorphismρ : Γ→ G is uniquely determined by (ρ(γ1), ...,ρ(γn)) ∈ Gn when γ1, ...,γn ∈ Γ isa set of generators. On the other hand, if we fix a presentation of Γ, then an n-tuple(g1, ...,gn) ∈ Gn will induce an element in Hom(Γ,G) whenever {gi}ni=1 satisfythe relations in the presentation of Γ. Thus, there is a one to one correspondencebetween the subset of such n-tuples in Gn and Hom(Γ,G). Topologize Hom(Γ,G)with the subspace topology on Gn.Lemma 1.1.1. Let ϕ : Γ→ Γ′ be a homomorphism of finitely generated groups. IfG is a topological group, then ϕ∗ : Hom(Γ′,G)→ Hom(Γ,G) is continuous.Proof. Suppose Γ= 〈a1, ...,ar |R〉 and Γ′= 〈b1, ...,bm |R′〉.Recall that the inducedmap ϕ∗ : Hom(Γ′,G)→ Hom(Γ,G) is given by(ρ(b1), ...,ρ(bm)) 7→ (ρ(ϕ(a1)), ...,ρ(ϕ(ar)))for ρ : Γ′→ G. For any i, ϕ(ai) = bni1i1 · · ·bniqiiqi. By fixing one presentation for eachϕ(ai) we get that ϕ∗ is given by(ρ(b1), ...,ρ(bm)) 7→(ρ(bn1111 · · ·bn1q11q1), ...,ρ(bnr1r1 · · ·bnrqrrqr )) =(ρ(b11)n11 · · ·ρ(b1q1 )n1q1 , ...,ρ(br1)nr1 · · ·ρ(brqr )nrqr ).Therefore ϕ∗ is the restriction of the map Gm→ Gr given by(g1, ...,gm) 7→ (gn1111 · · ·gn1q11q1, ...,gnr1r1 · · ·gnrqrrqr )which is continuous.In particular, this Lemma tells us that given any two presentations of Γ, we getan isomorphism ϕ : Γ→ Γ and hence a homeomorphism ϕ∗ between the inducedspaces of homomorphisms. Therefore the topology on the space of homomor-phisms does not depend on the choice of presentations.7Recall that an affine variety is the zero locus in kn of a family of polynomialson n variables over a field k. Throughout this paper we will focus only on k = R.An affine variety that has a group structure with group operations given bypolynomial maps, i.e., maps f = ( f1, ..., fn) where each fi is a polynomial, is calleda linear algebraic group. For example, consider any matrix group. It is easy tocheck that matrix multiplication is in fact a polynomial map. For the inverse op-eration of matrices, it is easier to think of matrix groups as subgroups of SL(n,R).Any matrix A in SL(n,R) satisfies A−1 =Ct , the transpose of the cofactor matrix Cof A. Since the cofactor matrix is described only in terms of minors of A, the mapA 7→ Ct is a polynomial map. In fact, this is the general example, since it can beshown that any linear algebraic group is isomorphic to a group of matrices (see forexample [20, p. 63]).Lemma 1.1.2. Let G be a linear algebraic group, then for any finitely generatedgroup Γ, Hom(Γ,G) is an affine variety. Moreover, if ϕ is a homomorphism offinitely generated groups, then ϕ∗ is a polynomial map.Proof. Suppose Γ is generated by γ1,γ2, ...,γr and has a presentation {pα}α∈Λ.Each pα is of the form γn1k1 · · ·γnqkq = e, n j ∈ Z and γkl ∈ {γ1, ...,γr} for all 1≤ j ≤ q.For any homomorphism ρ : Γ→ G and any such relation pα we haveρ(pα) = ρ(γn1i1 · · ·γnqiq ) = ρ(γi1)n1 · · ·ρ(γiq)nq = I,the identity matrix in G. Since products and inverses in G are given in terms ofpolynomials, this sets up a family of polynomial relations {yα,i, j}α,i, j, where eachyα,i, j is induced by ρ(pα)i, j = δi j, the i, j entry of the matrix equality ρ(pα) = I.These relations do not depend on ρ , only on pα , in the sense that any r-tuple(g1, ...,gr) ∈ G satisfying {yα,i, j}α,i, j, i.e.(gn1i1 · · ·gnqiq )i, j = δi jfor all α ∈ Λ and 1 ≤ i, j ≤ n, is an element of Hom(Γ,G). Adding the polyno-mial relations {yα,i, j}α∈Λ to the ones describing Gr define Hom(Γ,G) as an affinevariety.For the second part, recall from the proof of Lemma 1.1.1, that ϕ∗ is defined in8terms of products and inverses of matrices and thus a polynomial map.Similarly, this lemma tells us that the affine variety structure on Hom(Γ,G)does not depend on the presentation of Γ. Indeed, any isomorphism of groups willinduce an isomorphism of affine varieties.1.1.2 Triangulation of Semi-algebraic SetsDefinition 1.1.3. A real semi-algebraic set is a finite union of subsets of the form{x ∈ Rn | fi(x)> 0, g j(x) = 0 for all i, j},where fi(x) and g j(x) are a finite number of polynomials with real coefficients.Using Hilbert’s basis theorem, all affine varieties overR are real semi-algebraicsets. Indeed, the zero locus ideal of an affine variety will be finitely generated andthus the affine variety can be carved out by finitely many polynomials.An interesting property of real semi-algebraic sets is that their image under apolynomial map Rn→ Rm is also a semi-algebraic set in Rm (see [19, p. 167]), asopposed to affine varieties and regular maps.Let M, N be semi-algebraic subsets of Rm and Rn, respectively. A continuousmap f : M→ N is said to be semi-algebraic if its graph is a semi-algebraic set inRm×Rn. The next result is proven in [19, p. 170].Proposition 1.1.4. Given a finite system of bounded semi-algebraic sets Mi inRn, there is a simplicial complex K in Rn and a semi-algebraic homeomorphismk : |K| →⋃Mi where each Mi is a finite union of k(int|σ |)’s with σ ∈ K.Remark 1.1.5. Proposition 1.1.4 can be stated without the boundedness conditionand the details can be found in [26, Theorem 2.12], where they add the hypothesisthat⋃Mi is closed in Rn.In the next sections, we will be using this last result in its full extension, but afirst application is that any affine variety Z can be triangulated, that is, there existsa simplicial complex K and a homeomorphism |K| ∼= Z. With Lemma 1.1.2 andProposition 1.1.4 we prove the following.9Corollary 1.1.6. Let Γ be a finitely generated group and G a real linear algebraicgroup. Then Hom(Γ,G) is a triangulated space.1.1.3 Simplicial Spaces and Homotopy Stable DecompositionsLet ∆ be the category of finite sets [n] = {0,1, ...,n} with morphisms order pre-serving maps f : [n]→ [m]. It can be shown that all morphisms in this category aregenerated by composition of maps denoted di : [n−1]→ [n] and si : [n+1]→ [n]where 0≤ i≤ n. These maps are determined by the relationsd jdi = did j−1 if i < js jsi = si−1s j if i > js jdi =dis j−1 if i < jId if i = j or i = j+1di−1s j if i > j+1,which are called cosimplicial identities. For any category C, let Cop denote itsopposite category. A functorX : ∆op→ Topis called a simplicial space. Here Top stands for k-spaces, i.e., topological spaceswhere each compactly closed subset is closed. We denote Xn := X([n]) and themaps di = X(di) and si = X(si) are called face and degeneracy maps respectively.Fix n. Define S0(Xn) = Xn and for 0 < t ≤ nSt(Xn) =⋃Jn,tsi1 ◦ · · · ◦ sit (Xn−t),where si j : Xn− j→ Xn− j+1 is a degeneracy map, 1≤ i1 < · · ·< it ≤ n is a sequenceof t numbers between 1 and n, and Jn,t stands for all possible sequences. Thisdefines a decreasing filtration of Xn,Sn(Xn)⊂ Sn−1(Xn)⊂ ·· · ⊂ S0(Xn) = Xn.For each n there is a homotopy decomposition of ΣXn in terms of the quotient10spaces Sk(Xn)/Sk+1(Xn) with k ≤ n. To do this we need the following.Let A⊂ Z be topological spaces. Recall that (Z,A) is an NDR pair if there existcontinuous functionsh : Z× [0,1]→ Z, u : Z→ [0,1]such that the following conditions are satisfied:1. A = u−1(0),2. h(z,0) = z for all z ∈ Z,3. h(a, t) = a for all a ∈ A and all t ∈ [0,1], and4. h(z,1) ∈ A for all z ∈ u−1([0,1)).Examples of NDR pairs are pairs consisting of CW-complexes and subcomplexes.Indeed, if Z is a CW-complex and A⊂ Z a subcomplex, then the inclusion A ↪→ Zis a cofibration which is equivalent to a retraction X× I to A× I∪X×{0} relativeto A×{0}.When X is a simplicial space, we call X simplicially NDR if (St−1(Xn),St(Xn))is an NDR pair for every n and t ≥ 1. The following result can be found in [1,Theorem 1.6].Proposition 1.1.7. Let X be a simplicial space, and suppose X is simpliciallyNDR. Then for every n≥ 0 there is a natural homotopy equivalenceΘ(n) : ΣXn '∨0≤k≤nΣ(Sk(Xn)/Sk+1(Xn)).For each n, the map Θ(n) is natural with respect to morphisms of simplicialspaces, that is, natural transformations X → Y .1.1.4 Cosimplicial Groups, 1-cocycles and Hom(L,G)Definition 1.1.8. Let Grp denote the category of groups. A functor L : ∆→ Grpis called a cosimplicial group. The homomorphisms di = L(di) and si = L(si) arecalled coface and codegeneracy homomorphisms respectively. We say that L is afinitely generated cosimplicial group if each Ln is finitely generated.There are two canonical finitely generated cosimplicial groups that arise from11finitely generated free groups.Definition 1.1.9. Define F : ∆ → Grp as follows: set F0 = {e} and for n ≥ 1let Fn = 〈a1, ...,an〉, the free group on n generators. The coface homomorphismsdi : Fn−1→ Fn are given on the generators byd0(a j) = a j+1di(a j) =a j j < ia ja j+1 j = ia j+1 j > ifor 1≤ i≤ n−1dn(a j) = a j;and the codegeneracy homomorphisms si : Fn+1→ Fn bysi(a j) =a j j ≤ ie j = i+1a j−1 j > i+1for 0≤ i≤ n.Definition 1.1.10. Define F : ∆→ Grp as Fn := 〈a0, ...,an〉 for any n ≥ 0; co-face and codegeneracy homomorphisms di: Fn−1 → Fn and si : Fn+1 → Fn re-spectively, are given on the generators bydi(a j) ={a j j < ia j+1 j ≥ iand si(a j) ={a j j ≤ ia j−1 j > ifor all 0≤ i≤ n.Definition 1.1.11. We will say that a family of normal subgroups Kn ⊂ Fn is com-patible with F , if di(Kn−1)⊂ Kn and si(Kn+1)⊂ Kn for all n and all i. Similarly wedefine compatible families of F .12Given {Kn}n≥0 a compatible family with F , we get induced homomorphismsFn−1di //FnFn−1/Kn−1di // Fn/KnFn+1si //FnFn+1/Kn+1si // Fn/Kn.DefineF/K : ∆→Grpas (F/K)n = Fn/Kn with coface and codegeneracy maps the quotient homomor-phisms di and si respectively. This way F/K is a finitely generated cosimpli-cial group. Similarly, with a compatible family {Kn}n≥0 of F , we can defineF/K : ∆→Grp.Example 1.1.12. We describe two families of finitely generated cosimplicial groupsthat can be constructed using F/K and F/K through the commutator subgroup.• Let A be a group, define inductively Γ1(A) = A and Γq+1(A) = [Γq(A),A] forq > 1. The descending central series of A isΓq(A)E · · ·E Γ2(A)E Γ1(A) = A.Given a homomorphism of groups φ : A→ B, φ [a,a′] = [φ(a),φ(a′)] for all a,a′ inA, so thatφ(Γq(A))⊂ Γq(B).Taking A = Fn, and denoting Γqn := Γq(Fn), we see that the family of normalsubgroups {Γqn}n≥0 is compatible with di and si. Thus we can define F/Γq as(F/Γq)n = Fn/Γqn for all q and 1 ≤ i ≤ n. In particular, for q = 2, we obtainFn/Γ2n = Zn for all n≥ 0.• Another example using the commutator is the derived series of a group A:A(q) E · · ·E A(1) E A(0) = Awhere A(i+1)= [A(i),A(i)]. Again, φ(A(q))⊂B(q) for any homomorphism φ : A→B.Thus F/F(q), where (F/F(q))n = Fn/F(q)n defines a finitely generated cosimplicial13group.Similarly, F(q)n+1,Γqn+1 ⊂ Fn define compatible families of F and we obtain thefinitely generated cosimplicial groups F/Γq∗+1 and F/F(q)∗+1.Example 1.1.13. Here is one example of a cosimplicial group that does not comefrom a compatible family. Let L0 = Σ2 = 〈τ〉, L1 = Σ3 = 〈σ1,σ2〉 and define cofacehomomorphismsL0 d0 //d1 //L1as d0(τ) = σ2 and d1(τ) = σ1. The codegeneracy homomorphism s0 : L1→ L0 isgiven by s0(σ1) = s0(σ2) = τ . This defines a 1-truncated cosimplicial group whichwe denote by Σ2,3, that is, a functor Σ2,3 : ∆≤1→Grp. Here ∆≤1 stands for the fullsubcategory of ∆ with objects [0] and [1]. We can extend Σ2,3 to ∆ by using its leftKan extension.For our purposes we describe the second stage of this extension: We have thatL2 = 〈a,b,c | a2 = b2 = c2, aba = bab, aca = cac, bcb = cbc〉,coface homomorphismsL1 d0 //d1 //d2 //L2are given byd0(σ1) = a d1(σ1) = c d2(σ1) = cd0(σ2) = b d1(σ2) = b d2(σ2) = aand codegeneracy homomorphismsL1 L2s1oos0oo14bys0(a) = s0(c) = σ1 s1(a) = s1(b) = σ2s0(b) = σ2 s1(c) = σ1.Remark 1.1.14. The symmetric groups Σn can not be assembled all together asa cosimplicial group. This is because there are no surjective homomorphismsΣn → Σn−1 for n ≥ 5 to use as codegeneracy homomorphisms. Indeed, given ahomomorphism ϕ : Σn→ Σn−1, kerϕ is a normal subgroup of Σn, that is An or Σn.Thus the image of ϕ is either the identity element or a subgroup of order 2.We describe another method of constructing new cosimplicial groups that arisefrom a given one. To do this, we recall a concept that was originally introducedin [12, p. 284] to define cohomotopy groups (and pointed sets) for a cosimplicialgroup.Definition 1.1.15. Let L be a cosimplicial group. The elements b in L1 satisfyingd2(b)d0(b) = d1(b) (1.1)are called 1-cocycles of L. The set of 1-cocycles is denoted Z1(L).If b is a 1-cocycle, then applying s0 to equation 1.1, we obtain s0d2(b) = eand using the cosimplicial identities, d1s0(b) = e, which implies b ∈ kers0. Defineinductively bn ∈ Ln as bn+1 = dn+1(bn), where b1 := b. These elements will satisfyd2(bn)d0(bn) =d1(bn) and (1.2)bn ∈kers0 (1.3)for all n≥ 1. Given a 1-cocycle b we build a new cosimplicial group.Construction of Lb: Define Lb : ∆→ Grp as follows. For each n ≥ 0, Lbn :=F0 ∗Ln with codegeneracy homomorphisms sib := Id ∗ si, i≥ 0. The coface homo-morphisms are dib := Id ∗ di for i > 0. To define d0b consider the homomorphismkn : F0→ F0 ∗Ln given by kn(a0) = a0bn for all n≥ 0, then d0b := kn ∗d0. There is15a canonical inclusionιb : L ↪→ Lbinduced by the inclusions Ln ↪→ F0 ∗Ln.Example 1.1.16. • When b = e, Le = F0 ∗ L, where F0 represents the constantcosimplicial group with value F0.• Consider the finitely generated free cosimplicial group F . The codegeneracyhomomorphism s0 : F1→ F0 is the constant map and thus kers0 = F1 = 〈a1〉. Alsod1(a1) = a1a2 = d2(a1)d0(a1),and hence a1 ∈ Z1(F). Note that any other power of a1 will fail to satisfy thecocycle condition (1.1), that is Z1(F) = {e,a1}. Let F+ = Fa1 . We denote thecanonical inclusion as ι+ : F ↪→ F+. A similar argument shows that Z1(F/Γq) ={e,a1} for q > 2. We also denote (F/Γq)a1 = F/Γq+ and ι+ : F/Γq ↪→ F/Γq+.• Consider F/Γ2. As in the previous example, kers0 = 〈a1〉, but since F2/Γ22 =Z2 all powers of a1 will satisfy the cocycle condition, that is, Z1(F/Γ2) = Z. Thusfor each positive m ∈ Z we get non isomorphic cosimplicial groups (F/Γ2)m andinclusions ιm : F/Γ2 ↪→ (F/Γ2)m. When m = 1, we denote (F/Γ2)1 = F/Γ2+.•Consider Σ2,3 defined in Example 1.1.13. The product σ1σ2 ∈ (Σ2,3)1 satisfiesd2(σ1σ2)d0(σ1σ2) = caab = cb = d1(σ1σ2)and thus σ1σ2 is a 1-cocycle and we get the cosimplicial group Σσ1σ22,3 .Now we turn our attention to spaces of homomorphisms. For any topologicalgroup G, its underlying group structure defines the functorHomGrp( ,G) : Grpop→ Set.If L is a cosimplicial group, the composition of functors HomGrp( ,G)L which wedenote by Hom(L,G) defines a simplicial set. Whenever L is finitely generated, foreach n we can topologize Hom(Ln,G) in a way that the induced face and degener-acy maps are continuous. Therefore we get the simplicial space Hom(L,G) : ∆op→Top. We list some known simplicial spaces.16• Hom(F,G) = NG the nerve of G as a category with one object;• Hom(F+,G) = (EG)∗, Steenrod’s model for the total space of the univer-sal principal G-bundle p : EG→ BG, where p is induced by the simplicial mapι∗+ : Hom(F+,G)→ Hom(F,G);• Hom(F/Γq,G) = (B(q,G))∗ the underlying simplicial space of the classify-ing space B(q,G) defined in [4], [5] (for q = 2) and [7];• Hom(F/Γq+,G) = (E(q,G))∗ the underlying simplicial space of the totalspace of the universal bundle p : E(q,G)→ B(q,G) also defined in [4], [5] (forq = 2) and [7]. Again, p is induced by ι∗+ : Hom(F/Γq+,G)→ Hom(F/Γq,G);• Hom(F ,G) = NG, the nerve of the category G that has G as space of objectsand there is a unique morphism between any two objects.Remark 1.1.17. Consider the morphism of cosimplicial groups γ : F → F givenon generators by γn(ai) = ai−1a−1i , where γn : Fn → Fn. Let G be a topologicalgroup. The induced map γ : NG→ NG is the underlying simplicial map of Segal’sfat geometric realization model for the universal G-bundle. A detailed version ofthis can be seen in [16, p, 66].1.1.5 Homotopy Stable Decomposition of Hom(Ln,G)Lemma 1.1.18. Let G be a linear algebraic group and L a finitely generatedcosimplicial group. Let si : Ln+1→ Ln be a codegeneracy map. Then, the image ofsi := (si)∗ : Hom(Ln,G)→ Hom(Ln+1,G) is a subvariety for all 0≤ i≤ n.Proof. Suppose Ln+1 = 〈a1, ...,ar〉. The homomorphism si : Ln+1→ Ln is surjec-tive so that Ln ∼= Ln+1/kersi. We can describe kersi = 〈{bα}α∈Λ〉 where each bαis a fixed product of powers of generators ak. Let ρ : Ln+1 → G be a homomor-phism. Then (ρ(a1), ...,ρ(ar)) is in si(Hom(Ln,G)) if and only if ρ(bα) = I for allα ∈ Λ. That is, these r-tuples in Hom(Ln+1,G) are determined by the polynomialequations {ρ(bα) = I}α∈Λ and hence they build up an affine variety.For a cosimplicial group L, denote St(Lk,G) := St(Hom(Lk,G)).17Theorem 1.1.19. Let G be a real algebraic linear group, and L a finitely generatedcosimplicial group. Then for each n we have homotopy equivalencesΘ(n) : ΣHom(Ln,G)'∨0≤k≤nΣ(Sk(Ln,G)/Sk+1(Ln,G)).Proof. Fix n ≥ 0. Using Proposition 1.1.7, we only need to show that the pair ofspaces (St−1(Ln,G),St(Ln,G)) is a strong NDR-pair for all 0 ≤ t ≤ n. By Lemma1.1.18, each s j(Hom(Lk,G)) is an affine variety for all 0 ≤ j,k ≤ n. Then, for allt ≥ 1 the finite unionSt(Ln,G) =⋃Jn,tsi1 ◦ · · · ◦ sit (Hom(Ln−t ,G))is also an affine variety. Consider the natural filtrationSn(Ln,G)⊂ Sn−1(Ln,G)⊂ ·· · ⊂ S0(Ln,G) = Hom(Ln,G).The union⋃t St(Ln,G) = Hom(Ln,G) is an affine variety, and therefore is a closedsubspace of some euclidean space. By Remark 1.1.5, Hom(Ln,G) can be triangu-lated in a way that each St(Ln,G) is a finite union of interiors of simplices. SinceSt(Ln,G) are closed subspaces, it follows that under the triangulation they are sub-complexes. This way the inclusions St(Ln,G) ⊂ St−1(Ln,G) are cofibrations andhence NDR-pairs. Therefore Hom(L,G) is simplicially NDR.Lemma 1.1.20. Let G be a topological group and consider the cosimplicial groupF/Γq. ThenSk(Fn/Γqn,G)/Sk+1(Fn/Γqn,G)∼=(nk)∨Hom(Fk/Γqk ,G)/S1(Fk/Γqk ,G)for all 1≤ k ≤ n.Proof. Let 1≤ i1 < · · ·< in−k ≤ n. Consider the projectionsPi1,...,im : Gn→ Gn−k18given by (x1, ...,xn) 7→ (xi1 , ...,xin−k). We claim that the image of Hom(Fn/Γqn,G)under this projection lies on Hom(Fn−k/Γqn−k,G). Indeed, each projection Pi1,...,imis induced by the homomorphism ϕ : Fn−k → Fn given on generators as a j = ai j .Since ϕ(Γqn−k) ⊂ Γqn we get the homomorphism ϕ : Fn−k/Γqn−k → Fn/Γqn whichproves our claim. Assemble the restrictions of Pi1,...,im to Hom(Fn/Γqn,G) so thatwe build up a continuous mapηn : Hom(Fn/Γqn,G)→∏Jn,kHom(Fn−k/Γqn−k,G)given by(x1, ...,xn) 7→ {Pi1,...,im(x1, ...,xn)}(i1,...,in−k)∈Jn,kwhere Jn,k runs over all possible sequences of length n−k, 1≤ i1 < · · ·< in−k ≤ n.Since all sequences (i1, ..., in−k) ∈ Jn,k are disjoint, the restrictionηn|k : Sk(Fn/Γqn,G)→∨Jn,kHom(Fn−k/Γqn−k,G)/S1(Fk/Γqk ,G)has a continuous inverse∨Jn,k s j1 ◦ · · · ◦ s jk where 1 ≤ j1 < · · · < jk ≤ n and theintersection { j1, ..., jk} ∩ {i1, ..., in−k} = /0. Therefore ηn|k is a homeomorphismfor every k. Finally note that Sk+1(Fn/Kn,G) is mapped to∨S1(Fn−k/Γqn−k,G).Taking quotients we get the desired homeomorphism.The next corollary was first conjectured in [4, p. 12] for closed subgroups ofGLn(C). Since any real linear algebraic group is Zariski closed we have the fol-lowing version of the conjecture which follows from Theorem 1.1.19 and Lemma2.1.3.Corollary 1.1.21. If G is a Zariski closed subgroup of GLn(C), then there arehomotopy equivalences for the cosimplicial group F/Γq,Θ(n) : ΣHom(Fn/Γqn,G)'∨1≤k≤nΣ(nk)∨Hom(Fk/Γqk ,G)/S1(Fk/Γqk ,G)for all n and q.19Example 1.1.22. Let G = SU(2) and consider F/Γq.• The case q= 2 (Fn/Γ2n = Zn) has been largely studied (for example, [3], [10]and [14]). In this example we follow [3, pp. 482-484]. First a few preliminaries.Let T be the maximal torus of G that consists of all diagonal matrices(λ 00 λ)with λ ∈ S1 and W = N(T )/T = {[w],e} its Weyl group, where w =(0 −11 0).The group W acts on T via [w] · t = wtw−1 = t−1 and using left translation on G/Twe get a diagonal action on G/T ×T n. Let t∼= iR be the Lie algebra of T with theinduced action of W . There is an equivariant homeomorphism t→ T −{I} so thatG/T ×W (T −{I})n ∼=G/T ×W tn. The quotient map G/T → (G/T )/W ∼=RP2 is aprincipal W -bundle and we can take the associated vector bundle pn : G/T×W tn→RP2. Let λ2 be the canonical vector bundle over RP2, then we can identify pnwith nλ2, the Whitney sum of n copies of λ2. The pieces in the homotopy stabledecomposition before suspending areHom(Zn,SU(2))/S1(Zn,SU(2))∼={S3 if n = 1(RP2)nλ2/sn(RP2) if n≥ 2where (RP)nλ2 is the associated Thom space of nλ2 and sn is its zero section. There-foreΣHom(Zn,SU(2))' Σ∨nS3∨∨2≤k≤nΣ(nk)∨(RP2)kλ2/sk(RP2) .• Let q = 3. An n-tuple (g1, ...,gn) lies in Hom(Fn/Γ3n,SU(2)) if and onlyif [[gi,g j],gk] = I for all 1 ≤ i, j,k ≤ n, i.e., the commutators [gi,g j] are centralin the subgroup generated by g1, ...,gn. We claim that the center of every non-abelian subgroup of SU(2) lies inside {±I}. Suppose a,b are two elements inSU(2) such that [a,b] 6= I. Then, the cyclic groups 〈a〉 and 〈b〉 are contained indifferent tori T1 and T2, respectively. Since the center of 〈a,b〉 is abelian it mustlie in the intersection T1∩T2. These two circles can only intersect at {±I}, whichproves our claim. Therefore the central elements [gi,g j] are in {±I} for all 1 ≤20i, j ≤ n. ConsiderBn(SU(2),{±I}) = {(g1, ...,gn) ∈ SU(2)n | [gi,g j] ∈ {±I}}the space of almost commuting tuples in SU(2). By the previous observationHom(Fn/Γ3n,SU(2)) = Bn(SU(2),{±I}).In [3, pp. 485-486], it is shown thatBn(SU(2),{±I})/S1(SU(2),{±I})∼=Hom(Zn,SU(2))/S1(Zn,SU(2))∨K(n)PU(2)+where K(1) = 0 and for n ≥ 2, K(n) = 7n24 − 3n8 +112 . Here S1(SU(2),{±I}) arethe n-tuples in Bn(SU(2),{±I}) with at least one coordinate equal to I. SincePU(2)∼= RP3 and S1(SU(2),{±I}) = S1(Fn/Γ3n,SU(2)) we conclude thatΣHom(Fn/Γ3n,SU(2))' Σ∨nS3∨∨2≤k≤nΣ(nk)∨(RP2)kλ2/sk(RP2) ∨K(k)RP3+ .Remark 1.1.23. For q ≥ 4 we can find nilpotent subgroups of SU(2) of class q.Indeed, if ξn is a representative in SU(2) of a primitive n-th root of unity, then thesubgroup generated by the set {ξ2q,w} with w as above, is of nilpotency class q.With this we can show that the spaces Hom(Fn/Γqn,SU(2)) for q ≥ 4 have moreconnected components than Hom(Fn/Γ3n,SU(2)). More details in [8].1.1.6 Equivariant Homotopy Stable Decomposition of Hom(Ln,G)Let G,H be topological groups and f : G→ H a continuous homomorphism. If Lis a finitely generated cosimplicial group then for each n, we have the commutativediagramsHom(Ln,G)di //f∗Hom(Ln−1,G)f∗Hom(Ln,H)di // Hom(Ln−1,H),Hom(Ln,G)si //f∗Hom(Ln+1,G)f∗Hom(Ln,H)si // Hom(Ln+1,H)21for all n ≥ 0 and all 0 ≤ i ≤ n, so that f∗ is a simplicial map. Conjugation byelements of G defines a homomorphism G→ G so that Hom(Ln,G) is a G-spaceand each St(Ln,G) is a G-subspace.Definition 1.1.24. Let M be a G-space. We say that M has a G-CW-structure ifthere exists a pair (X ,ξ ) such that X is a G-CW-complex and ξ : X → M is a G-equivariant homeomorphism.We want to show that for all n, Hom(Ln,G) has a G-CW complex structure forwhich Sn(Ln,G)⊂ Sn−1(Ln,G)⊂ ·· · ⊂Hom(Ln,G) are G-subcomplexes. To showthis we slightly generalize some results in [26].We continue using the techniques of the previous section, so we require G to bea real linear algebraic group. It is known that any compact Lie group has a uniquealgebraic group structure (see [24, p. 247]). Assuming G is a compact Lie group,every representation space of G has finite orbit types (see [25]), so when M is analgebraic G-variety, the equivariant algebraic embedding theorem [26, Proposition3.2] implies that M has finite orbit types. Also, this theorem guarantees the exis-tence of a G-invariant algebraic map f : M→ Rd for some d such that the inducedmap f : M/G→ f (M) is a homeomorphism and f (M) is a closed semi-algebraicset in Rd ([26, Lemma 3.4]). If τ : |K| → M/G is a triangulation, we say thatτ is compatible with a family of subsets {Di} of M, if pi(Di) is a union of someτ(int|σ |), where σ ∈ K and pi : M→M/G is the quotient map.Proposition 1.1.25. Let G be a compact Lie group, M0 an algebraic G-varietyand {M j}nj=1 a finite system of G-subvarieties of M0. Then there exists a semi-algebraic triangulation τ : |K| →M/G compatible with the collection {M j(H) | His a subgroup of G}nj=0, where M j(H) = {x ∈M j | Gx = gHg−1 for some g ∈ G}.Proof. Let H1, ...,Hs ⊂ G be the orbit types of G on M and f : M→ Rd as above.By [26, Lemma 3.3] all M j(Hi) are semi-algebraic sets, and therefore all f (M j(Hi))are also semi-algebraic. Since i, j vary on finite sets, we can use Proposition 1.1.5and obtain a semi-algebraic triangulationλ : |K| → f (M) =⋃i jf (M j(Hi))22such that each f (M j(Hi)) is a finite union of some λ (int|σ |), where σ ∈ K. Takeτ = f−1 ◦λ .Proposition 1.1.26. Let G be a compact Lie group. Let M0 be an algebraic G-variety and {M j}kj=1 a finite system of G-subvarieties. Then M0 has a G-CW-complex structure such that each M j is a G-subcomplex of M.Proof. Let τ : |K| → M/G be as in Proposition 1.1.25 and pi : M→ M/G the or-bit map. Let K′ be a barycentric subdivision of K, which guarantees that forany simplex ∆n of K′, pi−1(τ(∆n − ∆n−1)) ⊂ M j(Hn) for some Hn ⊂ G and 0 ≤j ≤ k. Since τ| : pi−1(τ(∆n))/G→ ∆n is a homeomorphism and the orbit type ofpi−1(τ(∆n−∆n−1)) is constant, by [21, Lemma 4.4] there exists a continuous sec-tion s : τ(∆n)→M j so that s ◦ τ(∆n−∆n−1) has a constant isotropy subgroup Hn.Consequently there is an equivariant homeomorphismpi−1τ(∆n−∆n−1)∼= G/Hn× (∆n−∆n−1).Collecting G-cells Gs◦τ(∆n) for all simplices of K′ we get a G-CW structure overall M j, 0≤ j ≤ k.For a finitely generated cosimplicial group L, denoteRep(Ln,G) := Hom(Ln,G)/Gand St(Ln,G) := St(Ln,G)/G.Theorem 1.1.27. Let G be a compact Lie group and L a finitely generated cosim-plicial group. Then for each n, Θ(n) from Theorem 1.1.19 is a G-equivarianthomotopy equivalence, and in particular we get homotopy equivalencesΣRep(Ln,G)'∨1≤k≤nΣ(Sk(Ln,G)/Sk+1(Ln,G)).Proof. Assume G⊂GLN(R). Under conjugation by elements of G, Hom(Ln,G) isan affine G-variety and by Lemma 1.1.18 the subspaces S j(Ln,G) are G-subvarietiesfor all 1≤ j ≤ n. Hence, by Proposition 1.1.26 Hom(Ln,G) can be given a G-CW-23complex structure where each S j(Ln,G) is a G-subcomplex. Similarly, the quotientSk(Ln,G)/Sk+1(Ln,G) has a G-CW-complex structure.To prove that the map Θ(n) is a G-equivariant homotopy equivalence, firstrecall that conjugation by elements of G defines a simplicial action on Hom(L,G),and by the naturality of each Θ(n), the G-equivariance follows. Let H ⊂ G beclosed subgroup. The fixed point spaces Hom(Ln,G)H and Sk(Ln,G)H inherit aCW-complex structure so that Hom(L,G)H is simplicially NDR. By Proposition1.1.7 we have homotopy equivalencesΘ(n,H) : Σ(Hom(Ln,G)H)→∨0≤k≤nΣ(Sk(Ln,G)H/Sk+1(Ln,G)H)for each n ≥ 1. The fixed points map Θ(n)H agrees by naturality with Θ(n,H)and thus is a homotopy equivalence. The result now follows from the equivariantWhitehead Theorem.Corollary 1.1.28. Let G be a compact Lie group. Then the homotopy equivalencesin Corollary 1.1.21 are G-equivariant homotopy equivalences, and in particular wegetΣRep(Fn/Γqn,G)'∨1≤k≤nΣ(nk)∨Rep(Fk/Γqk ,G)/S1(Fk/Γqk ,G) .Example 1.1.29. Let G = SU(2) and L = F/Γq.• For q = 2, it was proven in [3, p. 484] thatRep(Zn,SU(2))/S1(Zn,SU(2))' T∧n/W = Sn/Σ2where the action of the generating element on Σ2 is given by(x0,x1, ...,xn) 7→ (x0,−x1, ...,−xn)for any (x0,x1, ...,xn). Identifying Sn = ΣSn−1, we can see the orbit space Sn/Σ2 as24first taking antipodes, and then suspending, that is Sn/Σ2 ∼= ΣRPn−1. ThusΣRep(Zn,SU(2))'∨1≤k≤nΣ(nk)∨ΣRPk−1 .• Let q = 3. We have shown that Rep(Fn/Γ3n,SU(2)) = Bn(SU(2),{±I})/Gand using the description of these spaces given in [3, p. 486], the stable pieces areRep(Fn/Γ3n,SU(2))/S1(Fn/Γ3n,SU(2))'∨K(n)S0∨ΣRPn−1.where K(n) is as in Example 1.1.22. ThereforeΣRep(Fn/Γ3n,SU(2))'∨1≤k≤nΣ(nk)∨∨K(k)S0∨ΣRPk−1 .1.2 Homotopy Properties of B(L,G)1.2.1 Geometric realization of Hom(L,G)Definition 1.2.1. Let L be a finitely generated cosimplicial group and G a topolog-ical group. DenoteB(L,G) := |Hom(L,G)|.For the cosimplicial groups F/Γq we get that B(F/Γq,G) = B(q,G), the clas-sifying space for G-bundles of transitional nilpotency class less than q. A naturalquestion is whether or not the space B(L,G) is a classifying space for a specificclass of G-bundles.Lemma 1.2.2. Let L be a cosimplicial group. The 1-cocycles of L are in one toone correspondence with cosimplicial morphisms F → L.Proof. Suppose b is a 1-cocycle. Any generator a j ∈ Fn is in the image of a1 ∈ F1under composition of coface homomorphisms, e.g., a j = (d0) j−1(d2)n− j(a1) for25all j ≥ 1. Define hn : Fn → Ln as hn(a j) = (d0) j−1(d2)n− j(b). To show that h iscosimplicial, consider the diagramsFn−1hn−1 //diLn−1diFnhn // LnFn+1hn+1 //siLn+1siFnhn // Ln.We prove the case of coface homomorphisms. Let a j ∈ Fn−1. On one side we gethndi(a j) =(d0) j−1(d2)n− j(b) j < i(d0) j−1(d2)n− j(b)(d0) j(d2)n− j−1(b) j = i(d0) j(d2)n− j−1(b) j > iand applying the cosimplicial identity dkdl = dldk−1 where k > l we obtaindihn−1(a j) =(d0) j−1(di− j+1)(d2)n− j−1(b) j < i(d0) j−1(d1)(d2)n− j−1(b) j = i(d0) j(d2)n− j−1(b) j > i.We need to analyze 2 cases:• j < i implies that i− j+1≥ 2, thus di− j+1(d2)n− j−1 = (d2)n− j.• j= i. The equality follows from equation 1.2 applied to (d2)n− j−1(b) = bn− j.Commutativity for the codegeneracy homomorphisms is similar, but using thecosimplicial identity skdl = dlsk−1 with k > l and condition 1.3 above. Hence h isuniquely determined by b. Given a morphism F→ L, the element b is given by theimage of a1 ∈ F1.Proposition 1.2.3. Let L be a cosimplicial group and hb : F→ L be the morphismdefined on F1→ L1 as a1 7→ b. Then the diagramF  ι+ //hbF+Id∗hbL  ιb // Lbis a pushout of cosimplicial groups.26Proof. Suppose f : F+→ K and g : L→ K are morphisms such that f ◦ ι = g◦hb.Define h : Lb→ K on each Lbn = F0 ∗Ln as hn(a0) = f n(a0) (here f n is evaluatedon a0 ∈ F0 ∗Fn) and hn(x) = gn(x) for any x ∈ Ln. To check that h is in fact acosimplicial homomorphism, by construction of Lb and h, we just need to verifycommutativity with coface maps at level i = 0. ConsiderLbn−1hn−1 //d0bKn−1d0Lbnhn // Kn.We only need to see what happens at a0 ∈ Lbn−1 :d0hn−1(a0) = d0 f n−1(a0) = f nd0+(a0) = fn(a0a1) = f n(a0) f n(a1) andhnd0+(a0) =hn(a0bn) = f n(a0)gn(bn).Denote b = b1. By hypothesis g1(b1) = f 1(a1). Sincegn(bn) =(d2)n−1g1(b1) andf n(a1) =(d2)n−1 f 1(a1),the desired equality holds.Corollary 1.2.4. Let G be a well based topological group and L a finitely gener-ated cosimplicial group. Using the notation above, suppose hb : F → L is a mor-phism. Then the inclusion ιb : L ↪→Lb defines a principal G-bundle |ι∗b | : B(Lb,G)→B(L,G).Proof. From the pushout diagram in Proposition 1.2.3, and applying the functorsHom( ,G) and geometric realization, we obtain the pullback diagramB(Lb,G) //|ι∗b |EGB(L,G)|h∗b| // BG27and hence |ι∗b | is a principal G-bundle.Example 1.2.5. We have seen that there is only one non-constant homomorphismha1 = Id : F → F . For q > 2 it can be shown that the same is true for L = F/Γq,where ha1 : F→F/Γq at each n is the quotient homomorphism. The correspondingB((F/Γq)+,G) is the space E(q,G) defined in [4, p. 94], and |h∗a1 | : B(q,G)→ BGis the inclusion. The bundle E(q,G)→ B(q,G) classifies transitionally nilpotentbundles of class less than q (see [7, section 5]). The case q = 2 is more in-teresting since Z1(F/Γ2) = Z. For m = 1 we obtain B((F/Γ2)+,G) = E(2,G)and E(2,G)→ B(2,G) classifies transitionally commutative bundles (see [5, sec-tion 2]). Since multiplication by −1 induces a cosimplicial automorphism ofF/Γ2, all constructions are equivalent for m = −1. Now let m > 1. The bun-dle B((F/Γ2)m,G)→ B(2,G) will classify G-bundles whose transition functionsgαβ : Uα ∩Uβ → G factor throughUα ∩Uβρα,β //gαβ##GmGwhere ραβ are transitionally commutative and m denotes taking the m-th power ofelements in G.1.2.2 Relation between commutative I-monoids and infinite loopspacesIn this section we recall briefly the notion of I-monoid and how it is related toinfinite loop spaces. This is more widely covered in [7]. Our goal is to use thismachinery to show that for a finitely generated cosimplicial group L, B(L,U) =colimnB(L,U(n)) is a non-unital E∞-ring space when Hom(L0,U) is path con-nected.Let I stand for the category whose objects are the sets [0] = /0 and [n] = {1, ...,n}for each n≥ 1, and morphisms are injective functions. Any morphism j : [n]→ [m]in I can be factored as a canonical inclusion [n] ↪→ [m] and a permutation σ ∈ Σm.This category is symmetric monoidal under two different operations, namely, con-28catenation [n]unionsq [m] = [n+m] with symmetry morphism the permutation τm,n ∈Σn+m defined asτm,n(i) ={n+ i if i≤ mi−m if i > mand identity object [0]. The second operation is Cartesian product [m]× [n] = [mn]with τ×m,n ∈ Σmn given byτ×m,n((i−1)n+ j) = ( j−1)m+ iwhere 1 ≤ i ≤ m and 1 ≤ j ≤ n. In this case the identity object is [1]. Cartesianproduct is distributive under concatenation (both left and right).Definition 1.2.6. An I-space is a functor X : I→ Top. This functor is determinedby the following.1. A family of spaces {X [n]}n≥0, where each X [n] is a Σn-space;2. Σn-equivariant structural maps jn : X [n]→ X [n+1] (here we consider X [n+1] is a Σn-space under the restriction of the Σn+1-action to the canonicalinclusion Σn ↪→ Σn+1) with the property: for any j : [n]→ [m], σ ,σ ′ ∈ Σmwhose restrictions in Σn are equal, we have σ · x = σ ′ · x ∈ X( j)(X [n]).We say that an I-space X is a commutative I-monoid if it is a symmetricmonoidal functor X : (I,unionsq, [0])→ (Top,×,{pt}). Additionally, we say that X isa commutative I-rig if X is also symmetric monoidal with respect to (I,×, [1]). Forthe latter definition we also require X to preserve distributivity.Definition 1.2.7. Let C be a small category and Y : C→ Top a functor. Denote byCnY the category of elements of Y , that is, objects are pairs (c,x) consisting of anobject c of C and a point x ∈ Y (c). A morphism in CnY from (c,x) to (c′,x′) is amorphism α : c→ c′ in C satisfying the equation Y (α)(x) = x′.Given Y : C→ Top, with the notation above, if we consider CnY as a topo-logical category whose space of objects and space of morphisms are⊔c∈obj(C)Y (c) and⊔f∈mor(C)Y ( f ),29respectively, then we have that the homotopy colimit of Y is the classifying spaceB(CnY ) = hocolimCY, that is, the realization of the nerve of the category CnY .Let X denote a commutative I-monoid. The category of elements InX is apermutative category, that is, a symmetric monoidal category where associativityholds by inspection. According to [23], the classifying space of a permutative cat-egory has an E∞-space structure, and so we get that hocolimIX has an E∞-spacestructure. Here we think of an E∞-space as a space with an operation that is asso-ciative and commutative up to a system of coherent homotopies. Thus, the groupcompletion ΩB(hocolimIX) is an infinite loop space. If X is a commutative I-rig,then InX is a bipermutative category and its classifying space is an E∞-ring space(as explained in [7]), that is, an E∞-space with an operation that is associative andcommutative (up to coherent homotopy) that is distributive (up to coherent homo-topy) over the E∞-space operation.Consider the subcategory of I consisting of the same set of objects and allisomorphisms. We denote it as P. The (bi) permutative structure on InX restrictsto Pn X , so that hocolimPX is also an E∞-space (E∞-ring space) and its groupcompletion ΩB(hocolimPX) is an infinite loop space (E∞-ring space). The mapsX [n]→∗ induce a map of (bi) permutative categories PnX → Pn∗ and thereforea map of infinite loop spaces (E∞-ring spaces)ρX : ΩB(hocolimPX)→ΩB(hocolimP∗).It follows that the homotopy fiber hofib ρX is an infinite loop space (non-unitalE∞-ring space). Denote X∞ := hocolimNX where N denotes the subcategory of Iwith same set of objects and as arrows the canonical inclusions, and X+∞ its Quillenplus construction applied with respect to the maximal perfect subgroup of pi1(X∞).The following proposition is proved in [7, Theorem 3.1].Proposition 1.2.8. Let X : I→ Top be a commutative I-monoid. Assume that• the action of Σ∞ on H∗(X∞) is trivial;• the inclusions induce natural isomorphisms pi0(X [n]) ' pi0(X∞) of finitelygenerated abelian groups with multiplication compatible with the Pontrjagin prod-uct and in the center of the homology Pontrjagin ring;• the commutator subgroup of pi1(X∞) is perfect (for each component) and X+∞30is abelian.Then hofib ρX ' X+∞ , and in particular X+∞ is an infinite loop space.Note that the last two conditions of the previous Proposition are satisfied wheneach X [n] is connected and X∞ is abelian. Under these hypothesis X∞ has an infiniteloop space structure.1.2.3 Non-unital E∞-ring space structure of B(L,U)Our first example and application of the machinery described in the previous sec-tion is showing the classical resultcolimmU(m) =Uhas an infinite loop space structure. This will allow us to establish non-unital E∞-ring space structures on our spaces of interest.First, we show that U( ) is a commutative I-rig. Recall that Σm ⊂ U(m) aspermutation matrices, so that U(m) has a Σm action. Consider the inclusionsim : U(m)→U(m+1)A 7→(A 00 1)which are continuous and preserve group structure. The maps im restrict to thecanonical inclusions Σm ↪→ Σm+1, thereforeim(σ ·A) = im(σ)im(A)im(σ)−1 = σ · im(A),where σ ∈ Σm, A ∈U(m) and on the right hand side σ ∈ Σm+1. Now let σ ,σ ′ ∈Σr, m < r and suppose both restrictions to the subset {1, ...,m} determine equalpermutations in Σm. Denote i = ir ◦ ir−1 ◦ · · · ◦ im. Then, for A ∈U(m),σ · i(A) =((σ |m)A(σ |−1m ) 00 Ir−m)=((σ ′|m)A(σ ′|−1m ) 00 Ir−m)= σ ′ · i(A).Therefore U( ) : I→ Top is a functor. This I-space has a commutative I-rig struc-ture as follows. Let ⊕m,n : U(m)×U(n)→ U(m+ n) denote the block sum of31matrices, which is a group homomorphism. The (m,n) shuffle map U(n+m)→U(n+m) is given by A 7→ τm,n ·A. We have the commutative diagramU(m)×U(n)τ⊕m,n // U(m+n)τm,nU(n)×U(m) ⊕n,m // U(m+n)where τ(A,B) = (B,A). Therefore U( ) is a commutative I-monoid. The othermonoidal structure is given by⊗m,n : U(m)×U(n)→U(mn) the tensor product ofmatrices. Indeed, by definition τ×m,n · ⊗m,n(A,B) = ⊗n,mτ(A,B), where A ∈U(m)and B ∈U(n). Since the image ⊕m,n(U(m)×U(n)) correspond to direct sum, thenassociativity, left and right distributivity over ⊗m,n hold.Now we check the conditions of Proposition 1.2.8: the action of Σm on U(m) ishomologically trivial since conjugation action on U(m) is trivial up to homotopy,U(m) being path connected. The inclusions im are cellular and hence U( )∞ 'Uand since U is an H-space under block sum of matrices, it is abelian. ThereforeU( )∞ 'U is an infinite loop space (non-unital E∞-ring space).Lemma 1.2.9. Let L be a finitely generated cosimplicial group and G,H real al-gebraic linear groups. Let p1 : G×H→G and p2 : G×H→H be the projections.ThenB(L, p1)×B(L, p2) : B(L,G×H)→ B(L,G)×B(L,H)is a natural homeomorphism.Proof. Since G×H is a direct product, p1 and p2 are continuous homomorphismand thereforep = (p1)∗× (p2)∗ : Hom(L,G×H)→ Hom(L,G)×Hom(L,H)is a simplicial map. Its easy to check that in fact p is a simplicial isomorphism.Both G and H being real algebraic, imply that Hom(Ln,G) and Hom(Ln,H) havea CW-complex structure, and therefore are k-spaces. By [22, Theorem 11.5] the32compositionB(L,G×H) |p| // Hom(L,G)×Hom(L,H)||pi1|×|pi2|// B(L,G)×B(L,H)is a natural homeomorphism where |pi1 ◦ p|× |pi2 ◦ p|= B(L, p1)×B(L, p2).Proposition 1.2.10. Let L be a finitely generated cosimplicial group, then B(L,U( ))is a commutative I-rig.Proof. Consider the I-rig U( ). Both the structural maps im and the action byelements of Σm are continuous group homomorphisms and hence B(L,U( )) =B(L, )U( ) is an I-space. Also, block sum of matrices and tensor product aretopological group morphisms so that with Lemma 1.2.9 we can defineµm,n = B(L,⊕m,n)◦ (B(L, p1)×B(L, p2))−1 andpim,n = B(L,⊗m,n)◦ (B(L, p1)×B(L, p2))−1,where p1 : U(m)×U(n)→ U(m) and p2 : U(m)×U(n)→ U(n) are the projec-tions. Let p′1 : U(n)×U(m)→U(n) and p′2 : U(n)×U(m)→U(m) denote alsoprojections. Notice thatτ ◦B(L, p′2)×B(L, p′1) = B(L, p1)×B(L, p2)◦B(L,τ)(where τ as before, is the symmetry morphism in Top). This implies that all prop-erties satisfied by ⊕m,n and ⊗m,n will be preserved by µm,n and pim,n.Theorem 1.2.11. Let L be a finitely generated cosimplicial group and supposethat the space Hom(L0,U(m)) is path connected for all m≥ 1. Then, B(L,U) is anon-unital E∞-ring space.Proof. By Proposition 1.2.10, B(L,U( )) is a commutative I-rig. It remains tocheck the conditions of Proposition 1.2.8. Note that the conjugation action of Σnis homologically trivial since it factors through conjugation action on U(m). Sinceall Hom(L0,U(m)) are path connected, |Hom(L,U(m))|= B(L,U(m)) is path con-nected for all m ≥ 1. The colimit B(L,U) is also an H-space under block sum ofmatrices, and therefore abelian.33Example 1.2.12. The property pi0(Hom(L0,U(m))) = 0 for all m ≥ 1 is satisfiedby the following cosimplicial groups:• L = F/Γq and L = F/F(q) since L0 = {e} in both cases.• L = F/Γq and L = F/F(q) since Hom(L0,U(m)) =U(m) in both cases.• Consider Σ2,3, and the cosimplicial morphism hσ1σ2 : F → Σ2,3. The imagehσ1σ2(F) defines a cosimplicial subgroup of Σ2,3, such that hσ1σ2(F)0 = {e}.Remark 1.2.13. The results in this section also apply to the groups SU and Sp.For SO and O the proofs are not exactly similar, but still true. The arguments usedin [7, Theorem 4.1] also apply in our case.34Chapter 2Nilpotent n-tuples in SU(2)One of the basic problems in spaces of homomorphisms is to compute their numberof connected components. In this chapter we mainly focus on the spaces consistingof homomorphisms from the finitely generated free q-nilpotent groups Fn/Γqn intothe Lie group SU(2). As noted in the previous chapter, the case q= 3 is the same ascomputing the space of almost commuting tuples of SU(2) which were completelydescribed in [3]. We generalize this for q-nilpotent tuples with q ≥ 4. We provethat all non-abelian nilpotent subgroups of SU(2) are conjugated to the quaterniongroup Q8 or to one of the generalized quaternions Q2q of order 2q. Using this weprove the following.Theorem 2.0.14. Let q≥ 3 and n≥ 2. ThenHom(Fn/Γqn,SU(2))∼= Hom(Zn,SU(2))unionsq⊔C(n,q)PU(2)whereC(n,q) =2n−2(2n−1)(2n−1−1)3+(2n−1)(2(q−3)(n−1)−1)22n−3.Theorem 2.0.14 has immediate consequences for SO(3) and U(2). The doublecovering map pi : SU(2)→ SO(3) induces a surjective mapHom(Fn/Γqn,SU(2))→ Hom(Fn/Γq−1n ,SO(3)),35which we can use to compute the connected components of Hom(Fn/Γqn,SO(3)).The case q = 2 was originally proved in [27].Corollary 2.0.15. Let q≥ 3, n≥ 2 and Hom(Zn,SO(3))1 be the connected com-ponent that contains the identity n-tuple (I, . . . , I). ThenHom(Fn/Γqn,SO(3))∼= Hom(Zn,SO(3))1unionsq⊔M(n,2)S3/Q8unionsq⊔M(n,q)S3/C4,where M(n,2) = (2n−1)(2n−1−1)3 , M(n,q) = (2n− 1)(2(q−3)(n−1)− 1)2n−2 and C4 isa cyclic group of order 4. Both C4 and Q8 act by conjugation on S3.The group homomorphism S1×SU(2)→U(2) given by (λ ,X) 7→ λX inducesa surjective map(S1)n×Hom(Fn/Γqn,SU(2))→ Hom(Fn/Γqn,U(2))and modding out the left hand side by the natural diagonal action of (Z/2)n we geta homeomorphism.Corollary 2.0.16. Let q≥ 3 and n≥ 2, then Hom(Fn/Γqn,U(2)) is homeomorphictoHom(Zn,U(2))unionsq⊔M˜(n,3)(S1)n×(Z/2)2 PU(2))unionsq⊔M˜(n,q)(S1)n×(Z/2) PU(2),where M˜(n,q) = M(n,q−1) is as in the previous Corollary.We also include a description of the space of representations Rep(Fn/Γ3,U(m))that expresses it as a union of almost commuting tuples of block matrix subgroupsof U(m) that can be identified with the direct product U(m1)×·· ·×U(mk) wherem1+ · · ·+mk = m.In section 2, using the homotopy stable decomposition obtained in Chapter 1for the spaces Hom(Fn/Γqn,G) (for G a real algebraic linear group), we compute thestable homotopy type of the spaces Hom(Fn/Γqn,SU(2)) and Hom(Fn/Γqn,SO(3))after one suspension.36Lastly, we describe the homotopy type of the classifying spaces B(q,SU(2)) asthe homotopy pushoutsPU(2)×Nq B(q−1,Q2q)// B(q−1,SU(2))PU(2)×Nq BQ2q // B(q,SU(2)),where Nq is the normalizer of the dihedral group D2q−1 in PU(2). In particu-lar, this shows that the inclusions B(q− 1,SU(2))→ B(q,SU(2)) are a rationalhomology isomorphism for any q ≥ 3. We also show two interesting principalBZ/2-bundles, namely B(q,SU(2))→ B(q− 1,SO(3)) and BS1×B(q,SU(2))→B(q,U(2)), where the maps are induced by the double coverings SU(2)→ SO(3)and S1×SU(2)→U(2).2.1 Non-abelian nilpotent subgroups of SU(2)Let T ⊂ SU(2) denote the maximal torusT ={(λ 00 λ)| λ ∈ C and |λ |= 1}and let w =(0 −11 0). A straightforward calculation shows:Lemma 2.1.1. Let x,y ∈ T . Then wx = xw and• [x,y] = 1;• [x,wy] = x2;• [wx,y] = y2;• [wx,wy] = x2y2.Recall that a group Q is nilpotent if Γq+1(Q) = 1 for some q. The least suchinteger q is called the nilpotency class of Q. Let ξn =(e2pii/n 00 e−2pii/n)∈ T be ann-th root of unity and let µn stand for the subgroup generated by ξn. For any q > 1,37the generalized quaternionsQ2q+1 := µ2q ∪wµ2qare of nilpotency class q. This follows from the fact that [ξ2n ,w] = ξ 22n = ξ2n−1 forall n≥ 1.Lemma 2.1.2. Let H ⊂ T ∪wT be a subgroup. Suppose that the r-th stage of itsdescending central series is Γr(H) = µ2n for some n > 0.1. If r > 2, then Γr−1(H) = µ2n+1 .2. If r = 2, then there exists t ∈ T such that tHt = Q2n+2 .Proof. We prove the case r = 2. Suppose Γ2(H) = [H,H] = µ2n . Since the com-mutator generates µ2n , there exist z0,wx0 ∈H, with x0 ∈ T such that [z0,wx0] = ξ k2nwith k odd (otherwise we’d have [H,H]⊆ µ2n−1). Let z ∈H∩T . Then [z,wx0] = z2is a power of ξ2n , that is, z is a power of ±ξ2n+1 and z ∈ µ2n+1 . Let wy ∈ H ∩wT ,where y ∈ T . The commutator [wy,wx0] = y2x02 = ξ p2n for some p > 0, thereforey = ±ξ p2n+1x0. Therefore H ⊆ µ2n+1 ∪wµ2n+1x0. To get the equality, consider theelement z0 ∈ H. We have 2 possibilities:• z0 is diagonal, and z20 = ξ2n implies z0 =±ξ2n+1 .• z0 is anti-diagonal, so that z0 = wz′0, with z′0 =±ξ2n+1x0.In both cases z0 and wx0z0 generate µ2n+1 ∪wµ2n+1x0. Conjugating any elementwξ p2n+1x0 by t =√x0 ∈ T we obtain√x0wξ p2n+1x0√x0 = w√x0ξ p2n+1x0√x0 = wξ p2n+1 .This is independent of the choice of the branch cut for√x0. Hence Γ1(tHt) =tHt ⊆ µ2n+1 ∪wµ2n+1 .For r > 2 the same arguments without the anti-diagonal matrices cases provethe result, since Γr−1(H)⊂ T .Lemma 2.1.3. Let X ,Y ∈ SU(2).1. If [X ,Y ] = I and g ∈ SU(2) diagonalizes X, then it also diagonalizes Y .382. If [X ,Y ] =−I and g∈ SU(2) diagonalizes X, then gXg−1 =±ξ4 and gY g−1 ∈wT .Proof. 2. Let g be a matrix that conjugates X to a diagonal matrix with eigenvaluesλ ,λ . Let X ′ = gXg−1 and Y ′ = gY g−1. Choose a non-zero vector v ∈ Eλ , theeigenspace of X associated to λ . Since [X ′,Y ′] =−I we getX ′Y ′v =−Y ′X ′v =−λY ′v.Hence −λ is an eigenvalue of X so that −λ = λ , which implies λ = ±i ∈ C.Therefore X ′ = ±ξ4. Now, v ∈ Ei and Y ′v ∈ E−i tells us that Y is conjugated by ginto an anti-diagonal matrix.Proposition 2.1.4. Let H ⊂ SU(2) be a nilpotent subgroup. Then either H isabelian or there exists a unique r ≥ 2 and an element g ∈ SU(2) such thatgHg−1 = Q2r+1 .Proof. Suppose H is not abelian. Then, there exists a unique r > 1 such thatΓr+1(H) = I and Γr(H) 6= I. This implies that Γr(H) lies inside the center ofH. Non-abelian subgroups of SU(2) have center contained in {±I} (see Example1.1.22) and therefore Γr(H) = [H,Γr−1(H)] = µ2. Fix an element X in Γr−1(H).For any h ∈ H, the commutator [X ,h] is inside {±I}. By Lemma 2.1.3, H canbe conjugated to T ∪wT by an element g ∈ SU(2) that diagonalizes X . Now,Γr(gHg−1) = µ2 and applying Lemma 2.1.2 inductively we get that there existst ∈ T , such that tgH(gt)−1 = Q2r+1 .Lemma 2.1.5. Let x,y ∈ T . Then• xyx−1 = y;• (wx)y(wx)−1 = y;• x(wy)x−1 = x2(wy) = x2yw = wx2y;• (wx)(wy)(wx)−1 = x2y2(wy) = x2y3w = wx2y3.Lemma 2.1.6. 1. Let q ≥ 3. The abelian subgroups of Q2q are all subgroups ofµ2q−1 or {±I,±wx} where x is an element of µ2q−1 .392. Let q≥ 4. The non-abelian subgroups of Q2q are of the form µ2r−1 ∪wxµ2r−1where x = (ξ2q−1)p with 0≤ p < 2q−r and 3≤ r ≤ q.Proof. 1. By Lemma 2.1.1, any subgroup containing an element of µ2q−1 −{±I}and one of wµ2q−1 has non-trivial commutator.2. By the proof of Lemma 2.1.2 any non-abelian nilpotent subgroup of T ∪wThas the form µ2n ∪wxµ2n for some x ∈ T . Therefore the non-abelian subgroups ofQ2q are µ2r−1 ∪wxµ2r−1 where x is not in µ2r−1 .Lemma 2.1.7. The normalizer of Q2q in SU(2) is:1. The Binary Octahedral group which is generated by{ξ8,w,12(1+ i −1+ i1+ i 1− i)}for q = 3;2. Q2q+1 for q > 3.Proof. For this it is more convenient to think of SU(2) as the group of unit quater-nions, which enables one to think geometrically in terms of rotations in the spaceof purely imaginary quaternions, a space we shall identify withR3. If q= cos(θ)+sin(θ)qˆ where qˆ is a unit length imaginary quaternion, then v 7→ qvq−1 maps imag-inary quaternions to imaginary quaternions and is a rotation around qˆ of angle2θ . An arbitrary quaternion can be written as a+ v where a is the real part andv is purely imaginary. We have q(a+w)q−1 = a+ qvq−1, so that conjugationby q preserves the real part and rotates the imaginary part. This rule that asso-ciates a rotation of R3 to each unit quaternion is the double cover homomorphismSU(2)→ SO(3).The normalizer of Q2q consists of those unit quaternions whose correspondingrotations are the orientation preserving symmetries of the set of imaginary partsof the elements of Q2q . As quaternions, the elements of Q2q are cos(2pik/2q−1)+isin(2pik/2q−1) and jcos(2pik/2q−1)+ k sin(2pik/2q−1) for k = 0,1, . . . ,2q−1− 1.The imaginary parts are then 2q−2 points on the i-axis together with the vertices ofa regular 2q−1-gon in the jk-plane. For q > 3, any orientation preserving isometryof R3 preserving this set of points must separately preserve the 2q−1-gon and the40points on the i-axis, and thus must be a rotation of R3 whose restriction to theplane of the 2q−1-gon is a symmetry of the 2q−1-gon. Each of the 2q elements ofthe dihedral group of symmetries of the 2q−1-gon extends to a unique rotation ofR3 (the rotations in the dihedral group extend to rotations around the i-axis, whilethe reflections extend to rotations of angle pi around the axis of the reflection), andeach such rotation of R3 has two quaternions inducing it; these 2q+1 quaternionsare easily seen to be the elements of Q2q+1 .Now, if q = 3, there is extra symmetry because the 2q−1-gon in the jk-plane isjust a square, and the 2q−2 = 2 points on the i-axis together with that square form aregular octahedron. Therefore the normalizer of Q23 is the preimage in SU(2) of thegroup of orientation preserving symmetries of the octahedron in SO(3); this preim-age is known as the Binary Octahedral group. The generator 12(1+ i −1+ i1+ i 1− i)listed in the statement of the proposition (which is the only generator not in Q24),corresponds to a rotation of 2pi/3 around a line connecting the centers of two op-posite faces of the octahedron. This extra symmetry does not separately preservethe i-axis and the square {± j,±k}.Remark 2.1.8. Here’s an alternative, matrix-based, proof of the second part. Letg ∈ NSU(2)(Q2q). Suppose gξ4g−1 = wx for some x ∈ µ2q−1 . Then gξ8g−1 is an ele-ment of order 8, and must lie in some subgroup of µ2q−1 , which is a contradiction.Therefore gξ4g−1 is a diagonal matrix and this can only be ±ξ4. By Lemma 2.1.3,g is in T ∪wT . Using Lemma 2.1.5, ξ2q ∈NSU(2)(Q2q) and for any g∈NSU(2)(Q2q),g2 is in Q2q , and thus NSU(2)(Q2q) = Q2q+1 .Theorem 2.1.9. Let q≥ 3 and n≥ 2. ThenHom(Fn/Γqn,SU(2))∼= Hom(Zn,SU(2))unionsq⊔C(n,q)PU(2)whereC(n,q) =2n−2(2n−1)(2n−1−1)3+(2n−1)(2(q−3)(n−1)−1)22n−3.Proof. Consider the map SU(2)× (Q2q)n → Hom(Fn/Γqn,SU(2)) given as conju-41gation by elements of SU(2). By Proposition 2.1.4 this map is surjective when werestrict to non-commuting tuples. Modding out by the center Z(SU(2)) = {±I} wehave the induced surjective mapψ : PU(2)× [(Q2q)n−Hom(Zn,Q2q)]→ Hom(Fn/Γqn,SU(2))−Hom(Zn,SU(2)).Fix an element x ∈ (Q2q)n−Hom(Zn,Q2q) and let g,h ∈ SU(2). Then gxg−1 =hxh−1 implies that gh−1 commutes with a diagonal matrix and with an anti-diagonalmatrix. Thus gh−1 = ±I. This shows that the map ψ is injective restricted toeach connected component. Now, let ∼ be the equivalence relation on (Q2q)n−Hom(Zn,Q2q) defined by: two elements are equivalent if they are conjugated toone another by some element in PU(2). We get the homeomorphismPU(2)×([(Q2q)n−Hom(Zn,Q2q)]/∼)∼=Hom(Fn/Γqn,SU(2))−Hom(Zn,SU(2)).Let Gen(n,Q2r)⊂ (Q2r)n be the subset of n-tuples that generate Q2r . The normal-izer NSU(2)(Q2r) acts on Gen(n,Q2r) by conjugation. The inclusions Gen(n,Q2r)⊂(Q2r)n induce a bijective functionq⊔r=3Gen(n,Q2r)/NSU(2)(Q2r)→ [(Q2q)n−Hom(Zn,Q2q)]/∼ .The action of NSU(2)(Q2r) on Gen(n,Q2r) is free once we take quotient by thecenter of the group, which acts trivially. Thus|[(Q2q)n−Hom(Zn,Q2q)]/∼ |=q∑r=3|Gen(n,Q2r)||NSU(2)(Q2r)|/2.By Lemma 2.1.7 we know the order of NSU(2)(Q2r). It remains to count the numberof elements in Gen(n,Q2r), for which we use an inclusion–exclusion argument: atuple generates Q2r if and only if its elements don’t all come from a single propermaximal subgroup of Q2r . So if M1,M2, . . . ,Mm are the maximal subgroups of Q2r ,we have|Gen(n,Q2r) = |Q2r |n−∑i|Mi|n+∑i, j|Mi∩M j|n−∑i, j,k|Mi∩M j ∩Mk|n+ · · ·42For Q8, there are three maximal subgroups, all isomorphic to Z/2×Z/2. theintersection of any two of them is the center of Q8, ±I. So we get|Gen(n,Q8)|= 8n−3 ·4n+3 ·2n−2n = 2n+1(2n−1)(2n−1−1).For r ≥ 4, it follows from 2.1.6 that things are pretty much the same: there arejust three maximal subgroups of Q2r , namely, µ2r−1 , µ2r−2 ∪wµ2r−2 and µ2r−2 ∪wξ22−1µ2r−2 . And the intersection of any pair of them is µ2r−2 , so we get|Gen(n,Q2r)|= (2r)n−3 · (2r−1)n+3 · (2r−2)n− (2r−2)n= 2(r−2)n+1(2n−1)(2n−1−1),and a quick calculation verifies the formula claimed.Remark 2.1.10. As noted in Example 1.1.22, the space Hom(Fn/Γ3n,SU(2)) isthe same as the space of almost commuting tuples Bn(SU(2),{±I}). In [3] theydescribe this space, and agrees completely with our computation for q = 3.2.1.1 Consequences for SO(3) and U(2)Theorem 2.1.9 can be used to describe the spaces of n-nilpotent tuples in SO(3) andU(2). We study first the ones in SO(3)∼= PU(2). Let pi : SU(2)→ SU(2)/{±I} ∼=PU(2) denote the quotient homomorphism, andpi∗ : Hom(Fn/Γqn,SU(2))→ Hom(Fn/Γqn,PU(2))the induced map. We claim that for any q > 2, the image of pi∗ is preciselyHom(Fn/Γq−1n ,PU(2)). Clearly any commuting tuple in SU(2) is mapped to acommuting tuple in PU(2). Now, let H be a non-abelian q-nilpotent subgroup ofSU(2). As showed before this implies that Γq(H)= {±I} and hence Γq(pi(H))= I.Therefore pi(H) has nilpotency class q− 1. It remains to show that any (q− 1)-nilpotent subgroup in PU(2) has a lift to a q-nilpotent subgroup of SU(2), but thisfollows from the fact that pi is an epimorphism and the preimage of any trivial43commutator is {±I}, and thus central in SU(2). Therefore, for any q≥ 3pi∗ : Hom(Fn/Γqn,SU(2))→ Hom(Fn/Γq−1n ,PU(2))is surjective. A result of W. M. Goldman [18, Lemma 2.2] implies that the restric-tion pi−1∗ (C)→ C to any connected component C of Hom(Fn/Γq−1n ,PU(2)), is a2n-fold covering map (the fiber is in correspondence with Hom(Fn/Γqn,{±I}n) ={±I}n). From Theorem 2.1.9 we know that the connected components of thespace Hom(Fn/Γqn,SU(2)) are either Hom(Zn,SU(2)) or Ki which denotes com-ponents homeomorphic to PU(2). Each of the latter components consist of conju-gated representations (by elements of PU(2)) of a fixed surjective homomorphismρ : Fn/Γqn → Q2r with 3 ≤ r ≤ q. The image under pi∗ of these components arethe conjugated homomorphisms (also by elements of PU(2)) of the correspondingepimorphism pi∗(ρ) : Fn/Γq−1n → D2r−1 . Since pi∗ is open, pi∗| : Ki→ pi(Ki) is alsoa covering map. Fix a representation σ : Fn/Γq−1n → D2r−1 in pi∗(Ki). Then thelifts of σ in Ki are gρg−1 : Fn/Γqn → gQ2r g−1 that once we mod out by {±I} weobtain ρ . Hence g is in Z(D2r−1). We have two different cases. First, if r ≥ 4,then Z(D2r−1) = 〈pi(ξ4)〉 and pi∗(Ki)∼= PU(2)/pi(C4), where C4 is the cyclic groupgenerated by ξ4. Therefore the restriction pi∗|Ki : Ki→ pi∗(Ki) is a 2-fold coveringmap. When r = 3, Z(D4) =D4, and pi|Ki : Ki→ pi∗(Ki)∼= PU(2)/pi(Q8) is a 4-foldcovering map.With these covering spaces, we can easily count the connected componentsof the space Hom(Fn/Γqn,SO(3)). Indeed, let Hom(Zn,SO(3))1 denote the con-nected component containing (I, ..., I). The connected component correspondingto Hom(Zn,SU(2)) is mapped under pi∗ to Hom(Zn,SO(3))1. For componentsof commuting tuples, other than the component Hom(Zn,SO(3))1, the correspon-dence is 2n4 to 1, and for components of r-nilpotent tuples with r > 2 it is2n2 to 1.Thus, we only need to divide the numbers C(n,3) and C(n,r)−C(n,3) of Theorem2.1.9 by 2n−2 and 2n−1 respectively.Corollary 2.1.11. Let q≥ 2, n≥ 2. ThenHom(Fn/Γqn,SO(3))∼= Hom(Zn,SO(3))1unionsq⊔M(n,2)S3/Q8unionsq⊔M(n,q)S3/C4,44where M(n,2) = (2n−1)(2n−1−1)3 , M(n,q) = (2n− 1)(2(q−3)(n−1)− 1)2n−2 and C4 isthe cyclic group of order 4 generated by ξ4. Both Q8 and C4 act by conjugation onS3.Remark 2.1.12. The case q = 2 was originally computed in [27], and the numberM(n,2) agrees with their calculation.Now we discuss the situation for U(2). Any matrix in X ∈U(2) can be writtenas√det(X)X ′, where X ′ ∈ SU(2). In this decomposition, [X ,Y ] = [X ′,Y ′] for anyX ,Y in U(2). Consider the map(S1)n×Hom(Fn/Γqn,SU(2))→ Hom(Fn/Γqn,U(2))given by (λ1, . . . ,λn,x1, . . . ,xn) 7→ (λ1x1, . . . ,λnxn). By the previous observationthis a surjective map. These representations are uniquely determined up to a nega-tive sign, that is, we have a homeomorphism(S1)n×(Z/2)n Hom(Fn/Γqn,SU(2))∼= Hom(Fn/Γqn,U(2)).Using Theorem 2.1.9 we get that the connected components of this space are home-omorphic to (S1)n×(Z/2)n Hom(Zn,SU(2)) ∼= Hom(Zn,U(2)) or (S1)n×H PU(2)where H is a subgroup of (Z/2)n. The first component corresponds to the com-muting n-tuples in U(2) and the later to the r-nilpotent tuples with 2 ≤ r ≤ q. Wecan see the (Z/2)n action as an action on the indexing set of the connected com-ponents, which can be represented by elements of Hom(Fn/Γqn,Q2q). To countthe number of connected components, let ~ε = (ε1, . . . ,εn) with each εi = ±1 bean arbitrary element in (Z/2)n and ~x = (x1, . . . ,xn) a non-commutative n-tuple inHom(Fn/Γqn,Q2q). Then the stabilizer of this element consists ofStab(~x) = {~ε | (ε1x1, . . . ,εnxn) = (gx1g−1, . . . ,gxng−1) for some g ∈ SU(2)}.That is, such g either commutes or anticommutes with all xi. We have several cases.Let~ε ∈ Stab(~x) be a non trivial element.Case 1: Suppose some xi lies in the torus T .45• If εi = 1, xi = gxig−1 and thus g must also lie in T . To generate an r-nilpotentsubgroup of Q2q with xi, we need at least one more element of the form ξ k2qw forsome k. This element does not commute with any element of T −{±I} and onlyanticommutes with ±ξ4. Thus, the only choices for g are ±I and ±ξ4.• If εi =−1, −xi = gxig−1 and by Lemma 2.1.3, xi =±ξ4 and g ∈ Tw. Again,in the n-tuple there must be an element of the form x j = ξ k2qw for some k. Theonly elements in Tw that commute or anticommute with x j are g = ±ξ k2qw org = ±ξ4ξ k2qw. Note that in this case, the remaining elements in the n-tuple canonly be of the form ±I,±ξ4,±ξ k2qw or ±ξ4ξ k2qw, which generates a copy of Q8.Case 2: Suppose all xi lie in Tw. Let xi = ξ k2qw.• Suppose ~x is an r-nilpotent tuple with r ≥ 3. Then there is at least one x j =ξ l2qw that is different from ±xi or ±ξ4xi. Thus the only choices for g are ±I and±ξ4.• Suppose~x is a 2-nilpotent tuple. Then the only other choices for the remain-ing x j’s are ±ξ k2qw or ±ξ4ξ k2qw. Hence g can only be ±I,±ξ4,±ξ k2qw or ±ξ4ξ k2qw.We can conclude that if ~x generates a copy of Q8, then |Stab(~x)| = 4 and|Stab(~x)|= 2 in any other case.Corollary 2.1.13. Let q≥ 3 and n≥ 2, then Hom(Fn/Γqn,U(2)) is homeomorphictoHom(Zn,U(2))unionsq⊔M˜(n,3)(S1)n×(Z/2)2 PU(2)unionsq⊔M˜(n,q)(S1)n×(Z/2) PU(2),where M˜(n,q) = M(n,q−1) is as in Corollary 2.1.11.Remark 2.1.14. It was proved in [11, Theorem 1] that given a finitely generatednilpotent group Γ, a complex reductive linear group G and K ⊂G a maximal com-pact subgroup, the inclusion induces a strong deformation retract Hom(Γ,G) 'Hom(Γ,K). Applying this to G = SL(2,C),SL(3,R) or GL(2,C) and to the com-pact subgroups K = SU(2),SO(3) or U(2) respectively, we also get a homotopytype description of Hom(Fn/Γqn,G) for all q≥ 2.462.1.2 2-nilpotent tuples in U(m)What about m > 2, what are the connected components of Hom(Fn/Γqn,U(m))then? We can not give an answer to this in its wide generality, but we can atleast say something about the case q = 3.Let a be a partition of {1,2, . . . ,m} into disjoint non-empty subsets. DefineU(a) as the subgroup of U(m) consisting of m×m “block diagonal matrices withblocks indexed by a”, by which we mean matrices A ∈U(m) whose (i, j)-th entryis 0 whenever i and j are in different parts of the partition a. To explain our ter-minology, notice that when each part of a consists of consecutive numbers, say, ifthe parts are {1, . . . ,m1}, {m1+1, . . . ,m1+m2}, {m1+m2+1, . . . ,m1+m2+m3},. . ., then A is what is traditionally called a block diagonal matrix:A =A1 0. . .0 Ak ; Ai ∈U(mi)The conjugacy class of the subgroup U(a) depends only on the sizes of theparts of a. To be specific, if pi is any permutation of {1, . . . ,m} such that the imageof each part of a consists of consecutive numbers, then U(a) is conjugate, via thepermutation matrix associated to pi , to the subgroup of traditional block diagonalmatrices as above (where the mi are the sizes of the parts of a). In particular, thesubgroup U(a) is always isomorphic to ∏ki=1U(mi) where the mi are the sizes ofthe parts of a but the isomorphism is not at all unique.Let Za denote the center of U(a) which consists of “block scalar matrices”:diagonal matrices diag(λ1, . . . ,λm) ∈U(m) such that λi = λ j whenever i and j arein the same part of a. For example, if the parts consist of consecutive numbers, theelements of Za are of the form:λ1Im1 0. . .0 λkImk .Given any diagonal matrix D = diag(λ1, . . . ,λm) ∈ U(m) there is a coarsest47partition a(D) such that D ∈ Za(D), namely, the partition where i and j are in thesame part if and only if λi = λ j. One can easily check that the centralizer of D isprecisely U(a(D)).Our goal is to interpret 2-nilpotent tuples of U(m) as almost commuting el-ements of the the subgroups U(a), that is, for any topological group G and anyclosed subgroup K ⊂ G contained in the center of G, the space of K-almost com-muting n-tuples is the subspace of Gn where each (x1, . . . ,xn) satisfies that [xi,x j]lies in K for all i, j. We denote this space as Bn(G,K) and we have an inclusionBn(G,Z(G))⊂ Hom(Fn/Γ3n,G).In particular, for U(m),⋃a`mBn(U(a),Za)⊂ Hom(Fn/Γ3n,U(m)),where we’ve borrowed the notation a ` m typically used for partitions of the num-ber m to indicate that the union is over all partitions of the set {1, . . . ,m} whereeach parts consists of consecutive numbers and the parts are ordered by size.Let Rep(Γ,G) denote the orbit space of the action of G on Hom(Γ,G) by con-jugation.Proposition 2.1.15. The above inclusion is surjective upon passing to orbits, thatis, (⋃a`mBn(U(a),Za))/U(m) = Rep(Fn/Γ3n,U(m)).Remark 2.1.16. The notation −/U(m) on the left is not meant suggest the unionis closed under the conjugation action! It just means the image of the union in theorbit space.Proof. Let (x1, . . . ,xn) be an element of Hom(Fn/Γ3n,U(m)). Then every commu-tator [xi,x j] is central in the group generated by {x1, . . . ,xn}. In particular, all com-mutators commute with each other. Since each xi is in U(m) and hence diagonaliz-able, we can simultaneously diagonalize all commutators by an element g ∈U(m).Let yi := gxig−1 and yi j := g[xi,x j]g−1. Now, each yi j lies in the center Zai j for somecoarsest partition ai j. Choose a as the infimum of all the ai j, that is, as the coarsest48partition refining all ai j. We have by construction yi j ∈ U(ai j) ⊆ U(a); and foreach k, we have that yk is in the centralizer of each yi j, so yk ∈⋂k U(ai j) =U(a).The last remaining detail is that this partition a may not have parts that consistof consecutive numbers, or those parts may not be ordered by size, but, as explainedabove, a further conjugation fixes that.Remark 2.1.17. The same argument works for SU(m) and its subgroups SU(a).There are only two minor differences: the first is that SU(a) is identified with thesubgroup of matrices in ∏ki=1U(mi) of determinant 1; the second is that for thelast bit of the proof, the “consecutivization”, one needs to observe that for everypartition one can always find an even permutation such that the image of each partconsists of consecutive numbers and permutation matrices for even permutationslie in SU(m).2.2 Stable Homotopy typeIn our previous description of Hom(Fn/Γqn,SU(2)), one of the connected compo-nents is Hom(Zn,SU(2)). In order to give a characterization of the homotopy typeof the space of nilpotent n-tuples we need a characterization of the commuting n-tuples in SU(2). Unfortunately this has not been yet obtained. Nevertheless, therecognition of the stable homotopy type has been achieved independently in [3],[10] and [14].Let S1(Fn/Γqn,SU(2)) denote the subspace of Hom(Fn/Γqn,SU(2)) consistingof n-tuples with at least one coordinate equal to I. In [3] they show thatHom(Zn,SU(2))/S1(Zn,SU(2))∼={S3 if n = 1(RP2)nλ2/sn(RP2) if n≥ 2(2.1)where (RP)nλ2 is the associated Thom space of nλ2, n times the Whitney sum ofthe universal bundle λ2 over RP2, and sn is its zero section. Using the homotopystable decomposition of the simplicial space induced by the commuting tuples in aclosed subgroup of GLn(C) (proved by the same authors in [4]) they get a completedescription of Hom(Zn,SU(2)) after one suspension.49In [10], they prove thatΣHom(Zn,SU(2))' Σ n∨k=1(nk)∨ΣS(kλ2)where S(kλ2) is the sphere bundle associated to kλ2. These two decompositionsagree since ΣS(kλ2)' (RP2)kλ2/sn(RP2).In M. C. Crabb’s paper [14], he expresses the stable homotopy type of the spaceHom(Zn,SU(2))+ as a wedge of various copies of RP2, RP4/RP2 and RP5/RP2.Now, by Theorem 1.1.27, if G is a compact Lie group, then there are G-equivariant homotopy equivalencesΣHom(Fn/Γqn,G)'∨1≤k≤nΣ(nk)∨Hom(Fk/Γqk ,G)/S1(Fk/Γqk ,G) (2.2)for all n and q. As a consequence of this and Theorem 2.1.9, we get the fol-lowing:Corollary 2.2.1. Let n≥ 1 and q≥ 3. There are homotopy equivalencesΣHom(Fn/Γqn,SU(2))'Σ∨nS3∨∨2≤k≤nΣ(nk)∨(RP2)kλ2/sk(RP2)∨ ∨K(k,q)RP3+whereK(n,q) =7n24− 3n8+112+q∑r=4(2r−1)n−3(2r−1−1)n+2(2r−2−1)n2r.Proof. For each q≥ 3, let H1,q(SU(2)) denote the complement of Hom(Zn,SU(2))in Hom(Fn/Γqn,SU(2)) and similarly, let S1,q(SU(2)) denote the complement ofS1(Zn,SU(2)) in S1(Fn/Γqn,SU(2)). We want to describe the stable pieces of thefiltration Hom(Fn/Γqn,SU(2))/S1(Fn/Γqn,SU(2)), which are homeomorphic toHom(Zn,SU(2))/S1(Zn,SU(2))∨H1,q(SU(2))/S1,q(SU(2)).50The right hand wedge sumand can be identified with the one point compactification(H1,q(SU(2))−S1,q(SU(2)))+ .Recall that in the proof of Theorem 2.1.9, H1,q(SU(2)) is shown to be⊔C(n,q)PU(2)by separating the non-abelian nilpotent n-tuples ~x according to the value of r forwhich the subgroup generated by~x is conjugate to Q2r . Since S1,q(SU(2)) consistsof non-commuting n-tuples with at least one entry equal to the identity I, we canuse similar reasoning but now for generating tuples without any coordinate equalto I, to obtain that the space H1,q(SU(2))−S1,q(SU(2))'⊔K(n,q)PU(2), whereK(n,3) =(23−1)n−3(22−1)n+2(2−1)n24=7n24− 3n8+112,and K(n,q) for q≥ 4 is as in the statement of the Corollary. Finally, the one pointcompactification(⊔K(n,q)PU(2))+=∨K(n,q)PU(2)+.. The remaining quotientspaces in (2.2) arise from commuting tuples, and these are given by (2.1). Theresult now follows.In [3] it also shown thatHom(Zn,SO(3))1/S1(Zn,SO(3))1 ∼={RP3 if n = 1(RP2)nλ2 if n≥ 2 (2.3)We use this to compute the homotopy stable decomposition of q-nilpotent tu-ples in SO(3).Corollary 2.2.2. Let n ≥ 1 and q ≥ 2. Then ΣHom(Fn/Γqn,SO(3)) is homotopyequivalent toΣ∨nRP3∨∨2≤k≤nΣ(nk)∨(RP2)kλ2 ∨ ∨N(k)(S3/Q8)+∨∨N(k,q)(S3/C4)+51whereN(n) =12(3n−1−1) and N(n,q) =q∑r=3(2r−1)n−3(2r−1−1)n+2(2r−2−1)n2n+r−2Proof. Consider the map pi∗ : Hom(Fn/Γq+1n ,SU(2))→ Hom(Fn/Γqn,SO(3)) as inthe proof of Corollary 2.1.11. Thenpi−1∗ (S1(Fn/Γqn,SO(3))) = S1(Fn/Γq+1n ,SU(2))∪S−1(Fn/Γq+1n ,SU(2))where the subscript −1 corresponds to the n-tuples with at least one coordinateequal to −I. By [18, Lemma 2.2], the restrictions pi∗|−1(C)→ C are coveringmaps for every connected component C, where pi∗| is the restriction to the sub-space Hom(Fn/Γq+1n ,SU(2))− pi−1∗ (S1(Fn/Γqn,SO(3))) whose image is the spaceHom(Fn/Γqn,SO(3))−S1(Fn/Γqn,SO(3)). Now we count the number of connectedcomponents ofE := Hom(Fn/Γq+1n ,SU(2))−pi−1∗ (S1(Fn/Γqn,SO(3)))which are the (q+ 1)-nilpotent n-tuples in SU(2) with no entries equal to I or−I. As before, the number of connected components of non-commuting nilpotentn-tuples of this subspace can be counted as(23−2)n−3(22−2)n+2(2−2)n24=2n(3n−1−1)8, for q = 3, andq∑r=4(2r−2)n−3(2r−1−2)n+2(2r−2−2)n2rfor q≥ 4.Recall that the correspondence of components of the total space E to the onesin the base space is 2n−2 to 1 for components of commuting tuples different ofHom(Zn,SU(2)) and 2n−1 to 1 for q> 2. Dividing the above numbers by 2n−2 and2n−1 respectively, we get the number of connected components ofHom(Fn/Γqn,SO(3))−S1(Fn/Γqn,SO(3)).52The one point compactification of this space yields all the nilpotent stable pieces in(2.2) different from Hom(Zn,SO(3))1/(Hom(Zn,SO(3))1∩S1(Zn,SO(3))), whichis given by (2.3). The result now follows from Corollary 2.1.11.Remark 2.2.3. For q = 3, the number of wedges homeomorphic to RP3+ in Corol-lary 2.2.1 agrees with the description made in [3] for ΣBn(SU(2),{±I}). Also, thestable decomposition of Hom(Zn,SO(3)) was first computed in [3] and the numberof wedges homeomorphic to (S3/Q3)+ in Corollary 2.2.2 agrees with N(n).Now we describe the spaces of representations Rep(Fn/Γqn,G) of our groups ofinterest. In [3] they show that Rep(Zn,SU(2))/((S1(Zn,SU(2))/SU(2)) ∼= Sn/Σ2where the action of the generating element in Σ2 is given by (x0,x1, . . . ,xn) 7→(x0,−x1, . . . ,−xn) for any (x0, . . . ,xn) in Sn. Identifying Sn with the suspensionΣSn−1, we can see the above action as first taking antipodes on Sn−1 and thensuspending, that is Sn/Σ2 = ΣRPn−1. ThereforeΣRep(Fn/Γqn,SU(2))'∨1≤k≤nΣ(nk)∨ ∨K(k,q)S0∨ΣRPk−1 .SimilarlyΣRep(Fn/Γqn,SO(3))'∨1≤k≤nΣ(nk)∨ ∨N(k)+N(k,q)S0∨ΣRPk−1 .For G =U(2), we use [3, Theorem 6.1, p. 473] to get thatRep(Zn,U(2))/(S1(Zn,U(2))/U(2))∼= (S1×S1)∧n/Σ2,where X∧n denotes the smash product of n copies of X and Σ2 acts by simultane-ously swapping the S1 factors in each smash factor (note that for a single smashfactor, S1×S1/Σ2 is a Mo¨bius band). Then, for q > 2,ΣRep(Fn/Γqn,U(2))'∨1≤k≤nΣ(nk)∨ ∨N(k)+N(k,q+1)S0∨ (S1×S1)∧k/Σ2 .532.3 Homotopy type of B(q,SU(2))We finish this chapter with a homotopy pushout description of B(q,SU(2)) in-troduced in section 1.2.1. Let q > 2 and consider the map PU(2)× (Q2q)n →Hom(Fn/Γqn,SU(2)) given by(g,x1, . . . ,xn) 7→ (gx1g−1, . . . ,gxng−1).It is well defined in the quotient PU(2)×Nq (Q2q)n→ Hom(Fn/Γqn,SU(2)), wherethe normalizer Nq := NPU(2)(D2q−1) acts by translation on PU(2) and by conju-gation on (Q2q)n. Consider the subsets Gen(n,Q2q),Hom(Fn/Γq−1n ,Q2q)⊂ (Q2q)nand the restrictions of the above map to the respective subspaces/0 //PU(2)×Nq Hom(Fn/Γq−1n ,Q2q) _// Hom(Fn/Γq−1n ,SU(2)) _PU(2)×Nq Gen(n,Q2q)  // PU(2)×Nq (Q2q)n // Hom(Fn/Γqn,SU(2)).We claim that all the above squares are pushouts. From the proof of Theorem 2.1.9we have the homeomorphismPU(2)×Gen(n,Q2q)/Nq ∼= Hom(Fn/Γqn,SU(2))−Hom(Fn/Γq−1n ,SU(2)).Since Gen(n,Q2q) is discrete, Nq is finite and acts freely, we have thatPU(2)×Gen(n,Q2q)/Nq ∼= PU(2)×Nq Gen(n,Q2q)which proves the outside square to be a pushout in sets. But, Hom(Fn/Γq−1n ,SU(2))is closed and PU(2)×Nq Gen(n,Q2q) is the image of a compact space, so thatHom(Fn/Γqn,SU(2)) has the disjoint union topology. This proves our claim forthe outside square, and a similar argument can be used for the square on the leftside. Now we look at the right square. The outside and left squares being pushoutsimply the right square is also a pushout. The middle arrow is a closed cofibra-tion, and thus the right square is also a homotopy pushout. Moreover, since themaps are either inclusions or given by conjugation, all arrows are simplicial maps.54Geometric realization commutes with colimits and homotopy colimits, and hencePU(2)×Nq B(q−1,Q2q)// B(q−1,SU(2))PU(2)×Nq BQ2q // B(q,SU(2))is a pushout of topological spaces and a homotopy pushout.Finally, here are some relations with the classifying spaces B(q,SO(3)) andB(q,U(2)). Let q ≥ 3 and consider the map pin := pi∗ : Hom(Fn/Γqn,SU(2))→Hom(Fn/Γq−1n ,SO(3)). In section 3 we said that pin is a 2n-fold covering map forevery n≥ 0. Moreover, we have the pullback diagramsHom(Fn/Γqn,SU(2))  //pinSU(2)n(pi)nHom(Fn/Γq−1n ,SO(3))  // SO(3)nand Hom(Zn,SU(2))  //pinSU(2)n(pi)nHom(Zn,SO(3))1  // SO(3)n.Again, all maps in the square are simplicial. Since geometric realization ofsimplicial spaces commutes with taking pullbacks, we obtain pullback squares oftopological spaces:B(q,SU(2)) //BSU(2)B(q−1,SO(3)) // BSO(3)and BcomSU(2) //BSU(2)BcomSO(3)1 // BSO(3),where BcomG = B(2,G). The right hand side arrow is a BZ/2-bundle, hence thetwo arrows on the left are also principal BZ/2-bundles.Now, let q≥ 2 and consider the group homomorphism ψ : S1×SU(2)→U(2)given by (λ ,x) 7→ λx, where S1 represents the scalar matrices in U(2). The kernelofψ is the group of order 2 generated by (−1,−I). For each n we have the pullback55square(S1)n×Hom(Fn/Γqn,SU(2))   //pin(S1×SU(2))n(ψ)nHom(Fn/Γqn,U(2))  // U(2)nwhich for the same reasons as before, after taking geometric realizations gives aBZ/2-bundleBZ/2→ BS1×B(q,SU(2))→ B(q,U(2)).56Chapter 3Cohomology RingsLet G be a compact and connected Lie group, T ⊂ G a maximal torus and W =N(T )/T its Weyl group. Recall that BcomG := |Hom(Z∗,G)|. In [5], the au-thors show that that there is an isomorphism in rational cohomology betweenH∗(BcomG1) and the invariants under the action of the Weyl group, (H∗(G/T )⊗H∗(BT ))W , and hence obtain the rational cohomology ring. The integral cohomol-ogy rings still remain unknown, and their presentations might not be the sameas in rational cohomology, since there is torsion in the cohomology groups ofBcomSU(2). In this chapter we give a presentation of the cohomology rings forG = SU(2) and U(2).Theorem 3.0.1. There is an isomorphism of graded ringsH∗(BcomU(2);Z)∼= Z[c1,c2,y1,y2]/(2y2− y1c1,y21,y1y2,y22)where ci ∈ H2i(BcomU(2);Z) and yi ∈ H2i+2(BcomU(2);Z). Moreover, ci is theimage of ci under the induced map of the inclusion BcomU(2)→ BU(2).Corollary 3.0.2. The map BcomSU(2)→ BcomU(2) arising from the inclusioninduces an epimorphism in cohomology rings with kernel generated by the class c1as in Theorem 3.1.3. In particularH∗(BcomSU(2);Z)∼= Z[c2,y1,x2]/(2x2,y21,x2y1,x22)57where c2,y1 ∈ H4(BcomSU(2);Z) and x2 ∈ H6(BcomSU(2);Z). The class c2 is theimage of c2 in H4(BSU(2);Z) under the inclusion BcomSU(2)→ BSU(2) in coho-mology.We also include a presentation for the ring H∗(BcomSO(3)1;Z), which showsthat the inclusion in integral cohomology H∗(BSO(3);Z)→ H∗(BcomSO(3)1;Z),is not injective.Using our classification of subgroups of the generalized quaternions made inChapter 2, and [4, Theorem 4.6] applied to Q2q we get thatB(r,Q2q)' B(Z/2q−1 ∗Z/2r−1 (∗2q−rZ/2r−1Q2r))where ∗A denotes the amalgamated product along A. The above formula allowed usto do some computations in cohomology for r = 2. In this case, the space B(2,G)is BcomG.Proposition 3.0.3. The cohomology ringH∗(BcomQ8;F2)∼= F2[y1,y2,y3,z]/(y21+ y22+ y23,yiy j, i 6= j),where yi has degree 1 and z degree 2. Let q≥ 4. ThenH∗(BcomQ2q ;F2)∼= F2[x1,x2,y1, . . . ,y2q−2 ,z]/(x21,xkyi,yiy j, i 6= j,x2+2q−2∑i=1y2i ),where x1,y j have degree 1 and x2,z have degree 2.Increasing the degree of nilpotency makes cohomology calculations consid-erably harder as we show it for the space B(3,Q16) (we verified our calculationsusing the computer algebra system SINGULAR as explained in the appendix).3.1 The space BcomG1The space BcomG was originally introduced in [4], and further studied in [5]. Thelatter authors among other results, they show that BcomG classifies principal G-bundles whose transition functions ραβ : Uα ∩Uβ →G are transitionally commuta-tive, that is, the functions ραβ commute whenever they are simultaneously defined.58In particular, the pullback of the universal bundle EG→ BG along the inclusionBcomG→ BG has a transitionally commutative structure. This bundle is denotedEcomG→ BcomG.When G is a compact and connected Lie group, a very useful tool towards theunderstanding of BcomG are the maximal tori T ⊂ G. Recall that Hom(Zn,G)1denotes the connected component of the trivial representation 1 : Zn → G in thespace Hom(Zn,G). By [9, Lemma 4.2], the map φ˜n : G×T n→Hom(Zn,G)1 givenby (g, t) 7→ gtg−1 is surjective. Moreover, φ˜n is invariant under the diagonal actionof N(T ) the normalizer of T , that is, by right translation on G and by conjugationon T n. Thus the induced mapφn : G/T ×W T n = G×N(T ) T n→ Hom(Zn,G)1is also surjective, where W = N(T )/T is the Weyl group of G. Since conjugationby elements of G is a group homomorphism, it is compatible with the simplicialstructure, therefore, after taking geometric realizations we obtainφ : G/T ×W BT → BcomG1,where BcomG1 := |Hom(Z∗,G)1|. It was proven in [2] that whenever any abeliangroup is contained in a path-connected abelian subgroup, then Hom(Zn,G) is con-nected. For example, this is true for G =U(n),SU(n) or Sp(n).3.1.1 Integral Cohomology ring of BcomSO(3)1We compute the cohomology ring for the case G = PU(2) that is isomorphic toSO(3). Let T be the maximal Torus of SU(2) consisting of the diagonal matrices.Recall that the Weyl group W acts on SU(2)/T by translation and on the classi-fying space BT by complex conjugation. Identifying SU(2)/T with S2, the actionof W is the action by antipodes. Thus, we have a fibration BT → S2×W BT →RP2.Cohomology of S2×W BT : We compute the integral cohomology of S2×W BTusing the Serre spectral sequence associated to the above fibration. To do this,first, we need to determine the action of pi1(RP2) on H∗(BT ;Z) ∼= Z[a], where59a ∈ H2(BT ;Z). This element corresponds to the generator in the cohomology ofthe first term of the filtration of BT , that is, ΣT ∼= S2. We describe the action inthis subspace. Let [ω] ∈ pi1(RP2) be the generator. The lift of ω is homotopyequivalent to the map ΣT → ΣT induced by complex conjugation T → T . Thismap in cohomology H1(T ;Z)→ H1(T ;Z) is multiplication by −1, and the sameholds for H2(ΣT ;Z) via the suspension isomorphism. This implies that the globalaction on H∗(BT ;Z) is generated by an 7→ (−a)n. We can conclude that the actionon Hq(BT ;Z) is the trivial representation of Z when q ≡ 0(mod 4) and the signrepresentation Zw when q≡ 2(mod 4).Now we can compute the E2 page of the Serre spectral sequence in cohomol-ogy. We have that E p,q2 = Hp(RP2;Hq(BT )), whereHq(BT ) =Z q≡ 0 (mod 4)Zw q≡ 2 (mod 4)0 else.Since H∗(RP2;Z) ∼= Z[u]/(2u,u2), where degu = 2, and by Poincare Duality,H p(RP2;Zw) = H2−p(RP2;Z), we get that the E∗,∗2 -page looks like.........8 a4Z 0 u4Z/27 0 0 06 0 x2Z/2 y2Z5 0 0 04 a2Z 0 u2Z/23 0 0 02 0 xZ/2 yZ1 0 0 00 Z 0 uZ/20 1 260and for p > 2, E p,q2 = 0. From this it follows that xy = uy = ux = y2 = x3 = 0.We claim that a2ny = y2n,a2nu = u2n,a2nx = x2n, for all n ≥ 1, and the class x2 inH2(RP2;H4(BT )) equals u2. Consider the diagramH p(RP2;Hq(BT ))×Hr(RP2;Hs(BT )) //H p+r(RP2;Hq+s(BT ))H p(RP2;Z/2)×Hr(RP2;Z/2) ^ // H p+r(RP2;Z/2)where the vertical arrows are induced by the reduction modulo 2 morphisms Zw→Z/2 and Z→ Z/2, respectively. These are well defined since both are pi1(RP2)-equivariant morphisms. Let ẑ ∈ H p(RP2;Z/2) be the reduction modulo 2 of z inH p(RP2;Hq(BT )). Since â2n = 1, the product a2nz with any generator z is notzero, which proves our claim except for x2. For this last class, notice that thebottom arrow maps the class x̂× x̂ to the generator in H2(RP2;Z/2), that is, x̂2. Bycommutativity of the diagram x2 is also a generator, and thus non-zero. This waywe obtain the relation x2−a2u = 0.All further differentials are zero and the spectral sequence collapses at the E2-page. Therefore, the E∞-page has the multiplicative structureE∗,∗∞ ∼= Z[b,u,x,y]/(2u,2x,u2,xy,uy,ux,y2,x2−bu,x3) (3.1)where degu = 2, degx = 3 and degb = degy = 4. It remains to show that allrelations between the generators still hold in H∗(S2×W BT ;Z), but this followsfrom the fact that all the “zero” relations take place in F p+qi , the i-th term ofthe associated filtration of H p+q(S2 ×W BT ;Z) with i ≥ 2. For i ≥ 3, they areall zero and F p+q2 is a subgroup. Therefore we have an isomorphism of gradedrings H∗(S2×W BT ;Z)∼= E∗,∗∞ .Let G = PU(2)∼= SU(2)/{±I} and T1 = T/{±I}. We claim that the squarePU(2)/N(T ) //PU(2)×N(T1) T n1φn∗ // Hom(Zn,PU(2))161is a homotopy pushout. Indeed, the restrictionφn| : PU(2)×N(T1) (T n1 −{I}n)→ Hom(Zn,PU(2))1−{(I, ..., I)}is injective since given t,s 6= (I, ..., I) such that gtg−1 = hsh−1, implies h−1g ∈N(T1) (two maximal tori K,K′ ⊂ PU(2) either intersect in I or K = K′). Moreover, since φn is surjective, φn| is a bijection. Thus the one point compactificationinduces a homeomorphism(PU(2)×N(T1) T n1 )/PU(2)/N(T1)∼= Hom(Zn,PU(2))1,and the square is a pushout of topological spaces. The diagonal action of N(T1) onPU(2)×T n1 induces a N(T1)-CW-complex structure such that PU(2) is an N(T1)-subcomplex. Therefore the inclusion PU(2)/N(T1) ↪→ PU(2)×N(T1) T n1 is a cofi-bration and the square is a homotopy pushout. Since geometric realization com-mutes with homotopy colimits, we obtain the homotopy pushoutPU(2)/N(T1) //PU(2)/T1×W BT1φ∗ // BcomPU(2)1,(3.2)using the identification PU(2)×N(T1) BT1 = PU(2)/T1×W BT1.Proposition 3.1.1. The cohomology ringH∗(BcomPU(2)1;Z)∼= Z[b,x,y]/(2x,y2,xy,x3),where degx = 3 and degb = degy = 4.Proof. The W action on PU(2)/T1 ∼= SU(2)/T ∼= S2 is by antipodes and on the cir-cles T and T1 is by complex conjugation, so that PU(2)/T1×W BT1 ∼= SU(2)/T×WBT . Using the above spectral sequence we can see that the projection S2×W BT1→RP2 is a cohomology isomorphism in degrees 0,1 and 2. Since the compositionRP2 ∼= PU(2)/N(T1)→ PU(2)/T1×W BT1→ RP2 is the identity map, the inclu-sion PU(2)/N(T1)→ PU(2)/T1×W BT1 is also a cohomology isomorphism in de-62grees 0,1 and 2. Applying the associated Mayer-Vietoris sequence to 3.2 we getthat the cohomology groups H i(BcomPU(2)1;Z) = 0 for i = 1,2, and that φ ∗ isan isomorphism for all i ≥ 3. The result now follows from the presentation of thecohomology ring 3.1.Remark 3.1.2. In [5, p. 521], the authors proved that the inclusion BcomG1→ BGis a monomorphism in rational cohomology. Proposition 3.1.1 shows that this isnot the case with integral coefficients. Indeed, H∗(BSO(3);Z) ∼= Z[χ, p1]/(2χ),where χ is the Euler class of the universal oriented bundle over BSO(3) (in degree3) and p1 is the first Pontrjagin class. The inclusion BcomSO(3)1→ BSO(3) sendsχ3 to zero and thus fails to be injective.3.1.2 Integral Cohomology ring of BcomU(2) and BcomSU(2)Consider the 2-fold covering map p : S1× SU(2)→U(2) (we think of S1 as thescalar matrices in U(2)). Let T2 ⊂U(2) denote the maximal torus consisting of thediagonal matrices in U(2). T2 is the image of S1×T under p, where T ⊂ SU(2) isthe subspace of diagonal matrices in SU(2). Notice that U(2) acts on S1×SU(2),trivially on S1 and by conjugation on SU(2), making p a U(2)-equivariant map.Let g ∈U(2) such that T2 6= g−1T2g and suppose x ∈ T2 ∩ g−1T2g. Let (λ ,x) bean element in p−1(x). Then x ∈ T ∩h−1T h = {±I} (where h√det(g) = g) whichimplies that x =±λ I, a scalar matrix. Since any conjugate of T contains the scalarmatrices, T2∩g−1T2g = S1.Now, consider φn : U(2)/T2×W T n2 → Hom(Zn,U(2)). The above reasoningimplies that the restrictionφn| : U(2)/T2×W (T n2 − (S1)n)→ Hom(Zn,U(2))− (S1)nis a bijection, and similarly as with PU(2) this induces the homotopy pushoutU(2)/N(T2)× (S1)n jn //pnU(2)×N(T2) T n2φn(S1)nin // Hom(Zn,U(2))63and taking geometric realization the squareRP2×BS1 j //pS2×W BT2φBS1 i // BcomU(2)(3.3)is also a homotopy pushout.Cohomology of S2×W BT2: Again, we have a fibration BT2 → S2×W BT2 →RP2. The action of the generating element in the Weyl group W on T2 is by swap-ping entries of the diagonal. Thus the induced action of pi1(RP2) on H2(BT2;Z) =〈a1,a2〉 is generated by (a1,a2) 7→ (a2,a1). This is the regular representation onZ[Z/2] = {m+ τn | m,n ∈ Z, τ2 = 1}. Since Hq(BT2) is zero for q odd and isgenerated by the elements ai11 ai22 , where i1, i2 ≥ 0 and i1 + i2 = q2 when q is even,we obtainHq(BT2) =Z[Z/2]n+1 q = 4n+2 n≥ 0Z[Z/2]n⊕Z q = 4n n≥ 00 else.Recall that the cohomology H p(RP2;Z[Z/2]) ∼= H p(S2;Z). Thus the E2-page ofthe Serre spectral sequence associated to the above fibration is given byE p,q2 = Hp(RP2;Hq(BT2))∼=Zn+1 p = 0,2 q = 4n+2 n≥ 0Zn+1 p = 0 q = 4n n≥ 0Zn⊕Z/2 p = 2 q = 4n n≥ 00 else.Consider the homomorphisms of Z[Z/2]-modules, Z[Z/2]→Z given by m+τn 7→m+n, whereZ is the trivial representation. By inspection of the cochain complexesand the induced morphism HomZ[Z/2](C∗(S2),Z[Z/2])→ HomZ[Z/2](C∗(S2),Z) itcan be shown that in homology, H2(RP2;Z[Z/2])→ H2(RP2;Z) is the reductionmodulo 2 homomorphism. Then, from the cohomology ring structure of RP2, it64follows that the multiplicative structure of the E∗,∗2 -page looks like.........4n+2 Zn+1 0 Zn+14n+1 0 0 04n Zn+1 0 Zn⊕Z/2............4 b21Z⊕b2Z 0 b1cZ⊕b2uZ/23 0 0 02 b1Z 0 cZ1 0 0 00 Z 0 uZ/20 1 2with c2 = u2 = b1u = cu = 0. Since Ep,q2∼= E p,q∞ and E p,q∞ = 0 for p ≥ 3, we haveno extension problems. This implies that the zero product in E∗,∗∞ are also zero inH∗(S2×W BT2;Z). HenceH∗(S2×W BT2;Z)∼= Z[b1,b2,c,u]/(2u,u2,c2,b1u,cu) (3.4)where degb1 = degu = 2 and degb2 = degc = 4.Now we study the homotopy pushout (3.3) in cohomology. Consider the trivialfibration BS1→ RP2×BS1→ RP2 and let E ′p,qr denote the associated Serre spec-tral sequence. The map j, induces a map of fibrations and hence a morphism ofspectral sequences j∞ : E p,q∞ → E ′p,q∞ . We use this map to see the values ofj∗ : H∗(S2×W BT2;Z)→ H∗(RP2×BS1)on our generators. Write H∗(RP2×BS1)∼= Z[u, t]/(2u,u2) with degu = deg t = 2.The map of fibrations is the identity map on base spaces, thus j∗(u) = j∞(u) =65u. We analyze the class j∞(c) ∈ H2(RP2;H2(BS1)). The map j∞ in bi-degree(2,2) is induced by the Z[Z/2]-equivariant homomorphism Z[Z/2] → Z givenby addition as abelian groups. The corresponding cohomology homomorphismH2(RP2;Z[Z/2])→ H2(RP2;Z) is reduction modulo 2. Both spectral sequencesconcentrate in degrees p = 0,2 which means that E2,2∞ and E′2,2∞ are actual sub-groups of H4(S2×w BT2) and H4(RP2×BS1) respectively. Thus j∗(c) = tu. Thegenerators bi ∈E0,2i∞ =H2i(S2×W BT2)/E2,1+(−1)i∞ require a more precise definitionand j∞ does not reveal enough information because the images j∞(bi) ∈ E ′0,2i∞ =H2i(RP2×BS1)/E ′2,1+(−1)i∞ for i = 1,2. To get canonical representatives of bi inH2i(S2×W BT2), consider the commutative diagramRP2×BS1 j //pS2×W BT2 φ // BcomU(2) _ιBS1  i1 // BT2  i2 // BU(2).Recall that T2 ↪→ U(2) induces an isomorphism H∗(BU(2);Z) ∼= H∗(BT2;Z)Wgiven by i∗2(c1) = a1+a2 and i∗2(c2) = a1a2, where ci is the i-th Chern class. Theni∗1(ak) = t implies that that (i2 ◦ i1 ◦ p)∗(c1) = 2t and (i2 ◦ i1 ◦ p)∗(c2) = t2. Defineb1 := (ι ◦φ)∗(c1) and b2 := (ι ◦φ)∗(c2).Note that any other representative of b1, say b1 + ku, still satisfies (b1 + ku)u = 0so that the presentation of the ring 3.4 remains unaltered. Now we have a completedescription of j∗, since by construction j∗(b1) = 2t and j∗(b2) = t2. It follows that( j∗− p∗)n : Hn(BS1)⊕Hn(S2×W BT2)→ Hn(RP2×BS1) is surjective for everyn≥ 0. Applying the Mayer-Vietoris sequence associated to (3.3) we obtain that thering homomorphismH∗(BcomU(2);Z)(i∗,φ∗)// H∗(BS1∨ (S2×W BT2);Z)∼= Z[t,b1,b2,c,u]/(2u,u2,c2,b1u,cu, tz; z ∈ {b1,b2,c,u})is injective. Therefore, H∗(BcomU(2);Z) is isomorphic to the subring im(i∗,φ ∗),66where im(i∗,φ ∗)n = ker( j∗− p∗)n for each n.Theorem 3.1.3. There is an isomorphism of graded ringsH∗(BcomU(2);Z)∼= Z[c1,c2,y1,y2]/(2y2− y1c1,y21,y1y2,y22)where ci ∈ H2i(BcomU(2);Z) and yi ∈ H2i+2(BcomU(2);Z). Moreover, ci is theimage of ci under the induced map of the inclusion BcomU(2)→ BU(2).Proof. Notice that the above reasoning implies that im(i∗,φ ∗) is generated by thekernel ker j∗ = (2c,b1c,b21−4b2) and the subring generated by {2t +b1, t2+b2}.Define the ring map f : Z[c1,c2,y1,y2] → im(i∗,φ ∗) on generators by f (c1) =2t + b1, f (c2) = t2 + b2, f (y1) = 2c and f (y2) = b1c. We claim that f is surjec-tive. Indeed, since b21−4b2 = f (c21−4c2) and all its non-zero multiples by genera-tors, b1(a21−4a2) = f (c31−4c1c2),a2(a21−4a2) = f (c21c2−4c22) and c(a21−4a2) =f (y2c1−2y2c2), our claim follows. Using SINGULAR (see A.1) we can verify thatthe kernel of f is (2y2− y1c1,y21,y1y2,y22). The last statement of the Theorem fol-lows from that fact that φ ∗ is injective in this case, and by the definitions of b1 andb2.Corollary 3.1.4. The map BcomSU(2)→ BcomU(2) arising from the inclusioninduces an epimorphism in cohomology rings with kernel generated by the class c1as in Theorem 3.1.3. In particularH∗(BcomSU(2);Z)∼= Z[c2,y1,x2]/(2x2,y21,x2y1,x22)where c2,y1 ∈ H4(BcomSU(2);Z) and x2 ∈ H6(BcomSU(2);Z). The class c2 is theimage of c2 in H4(BSU(2);Z) under the inclusion BcomSU(2)→ BSU(2) in coho-mology.Proof. In [17], the author verifies the existence of a homotopy fiber sequenceBcomSU(2)→ BcomU(2)→ BS1,and hence a homotopy fiber sequence S1 → BcomSU(2)→ BcomU(2). The E p,q2 -page of the associated Serre spectral sequence is concentrated in q = 0,1. Let67e be the generator of H1(S1;Z), and consider d2 : E0,12 → E2,02 . Then d2(e) =c1 ∈ H2(BcomU(2);Z), since BcomSU(2) is 3-connected (see [5]) and in particu-lar its cohomology in degrees 1 and 2 is zero. Therefore d2 : E4,12 → E6,02 satisfiesd2(ey1) = 2y2. Computing the remaining differentials d2, we obtain that the Ep,q3 -page is concentrated in degree q= 0, and is generated by c2,y1 and x2, the generatorof y2Z/2y2Z in bi-degree (6,0).Remark 3.1.5. The integral cohomology of BcomSU(2)was computed in [5], yield-ingH i(BcomSU(2);Z)∼=Z i = 0Z⊕Z i≡ 0 (mod 4), i > 0Z/2 i≡ 2 (mod 4), i > 20 else,which agrees with our computations. All our homotopy pushout diagrams involv-ing BcomG1, can also be derived from a more general result they prove, namely [5,Theorem 6.3].3.2 F2-Cohomology ring of BcomQ2qWe finish this chapter by showing that increasing the nilpotency class, even in finitegroups, has a meaningful impact in cohomology. In chapter 2, we have completelydescribed the subgroups of Q2q . This will allow us to compute the homotopy typeof B(r,Q2q) as follows. Let G be a finite group. Consider the categoryPr(G) withset of objects {Mα}∪{Mα ∩Mβ} where Mα are the maximal subgroups of G ofnilpotency class at most r. The set of arrows in Pr(G) is given by identities andinclusions. It was proved in [4] that when Pr(G) is a tree, there is a homotopyequivalenceB(r,G)' B(colimA∈Nr(G)A).Let q≥ 4. By Lemma 2.1.6 we conclude for 2≤ r < qB(r,Q2q)' B(Z/2q−1 ∗Z/2r−1 (∗2q−rZ/2r−1Q2r))68where ∗ denotes the amalgamated product of groups, and we let Q4 = Z/4.Cohomology of BcomQ2q: Taking r = 2, we get thatBcomQ2q ' B(Z/2q−1 ∗Z/2 (∗2q−2Z/2 Z/4))for any q ≥ 3. Applying the associated Mayer-Vietoris sequence inductively weobtainHn(BcomQ2q ;Z)∼=Z n = 0Z/2q−1⊕ (Z/2)2q−2 n even0 otherwise.Proposition 3.2.1. The cohomology ringH∗(BcomQ8;F2)∼= F2[y1,y2,y3,z]/(y21+ y22+ y23,yiy j, i 6= j),where yi has degree 1 and z degree 2. Let q≥ 4. ThenH∗(BcomQ2q ;F2)∼= F2[x1,x2,y1, . . . ,y2q−2 ,z]/(x21,xkyi,yiy j, i 6= j,x2+2q−2∑i=1y2i ),where x1,y j have degree 1 and x2,z have degree 2.Proof. We work out the case q≥ 4. Let Γ= Z/2q−1 ∗Z/2 (∗2q−2Z/2 Z/4). We use thecentral extensionZ/2C Γ→ Z/2q−2 ∗ (∗2q−2Z/2).Recall that the cohomology rings for n > 1, H∗(Z/2n;F2) ∼= F2[x1,x2]/x21 wheredeg(x1) = 1 and deg(x2) = 2. Thus,H∗(Z/2q−2 ∗ (∗2q−2Z/2);F2)∼= F2[x1,x2,y1, . . . ,y2q−2 ]/(x21,xkyi,yiy j, i 6= j).The k-invariant of the associated Serre spectral sequence of this extension is x2 +∑2q−2i=1 y2i . Therefore the E∗,∗3 page isF2[z]⊗F2[x1,x2,y1, . . . ,y2q−2 ]/(x21,xkyi,yiy j, i 6= j,x2+2q−2∑i=1y2i ).69The Steenrod square Sq1(x2 +∑2q−2i=1 y2i ) = Sq1(x2) is 0 since it can be expressedonly in terms of x2. That is, d3 = 0 and thus the spectral sequence abuts to E∗,∗3 .So we have found the E∞-page along with its ring structure. Since all therelations involve only generators from the base of the fibration, these relations holdin the cohomology ring as well.Remark 3.2.2. Recall that H∗(Q8;F2) = F2[x,y, t]/(x2+xy+y2,x2y+xy2) wherex,y have degree 1 and t has degree 4. Consider the inclusion ι : BcomQ8 → BQ8and ι∗ : H∗(BQ8;F2)→ H∗(BcomQ8;F2). By Proposition 3.2.1, ι∗(x2y) = 0 sinceall elements of degree 3 in the ring H∗(BcomQ8;F2) are multiples of the class z.Thus ι∗ is not injective.70Bibliography[1] A. Adem, A. Bahri, M. Bendersky, F. Cohen, and S. Gitler. “On decomposing sus-pensions of simplicial spaces”, Boletin Sociedad Matematica Mexicana (3) Vol.15(2009), 91-102.[2] A. Adem and F. Cohen, “Commuting elements and spaces of homomorphisms”,Mathematische Annalen 338 (2007), 587-626.[3] A. Adem, F. Cohen and J. M. Go´mez. “Stable splittings, spaces of representationsand almost commuting elements in Lie groups”, Math. Proc. Camb. Phil. Soc. Vol.149 (2010), 455-490.[4] A. Adem, F. Cohen and E. Torres-Giese. “Commuting elements, simplicial spacesand filtrations of classifying spaces”, Math. Proc. Cambridge Philos. Soc. Vol. 152(2012), 1, 91-114.[5] A. Adem and J. M. Go´mez. “A classifying space for commutativity in Lie groups”,Algebraic and Geometric Topology 15 (2015) 493-535.[6] A. Adem and J. M. Go´mez. “On the Structure of Spaces of Commuting Elementsin Compact Lie Groups”, Configuration Spaces: Geometry, Topology and Combina-torics. Publ. Scuola Normale Superiore, Vol. 14 (CRM Series), 2012, pp. 1-26.[7] A. Adem, J. M. Go´mez, J. Lind and U. Tillman. “Infinite loop spaces and nilpotentK-theory”, to appear in Algebraic and Geometric Topology.[8] O. Antolı´n Camarena and B. Villarreal, “Nilpotent n-tuples in SU(2)”,arxiv:1611.05937 Preprint 2017.[9] T. J. Baird. “Cohomology of the space of commuting n-tuples in a compact Liegroup”, Algebraic and Geometric Topology 7 (2007) 737-754.[10] T. Baird, L. Jeffrey, P. Selick. “The space of commuting n-tuples in SU(2)”, IllinoisJournal of Mathematics, Volume 55, Number 3, Fall 2011, Pages 805-813.[11] M. Bergeron. “The Topology of Nilpotent Representations in Reductive Groups andtheir Maximal Compact Subgroups”, Geometry and Topology 19-3 (2015), 1383–1407.71[12] A. K. Bousfield and D. M. Kan. “Homotopy limits, completions and localizations”,Lecture notes in mathematics, Vol. 304, Springer, Berlin (1972).[13] F. Cohen and M. Stafa. “On spaces of commuting elements in Lie groups”, Math.Proc. Camb. Phil. Soc. (2016), 161, 381-407.[14] M. C. Crabb. “Spaces of commuting elements in SU(2)”, Proceedings of the Edin-burgh Mathematical Society (2011) 54, 67-75.[15] W. Decker, G.-M. Greuel, G. Pfister, H. Scho¨nemann. SINGULAR 4-1-0 — A com-puter algebra system for polynomial computations. http://www.singular.uni-kl.de (2016). Retrieved March 2017.[16] J. L. Dupont. “Curvature and characteristic classes”, Lecture notes in mathematics,Vol 640, Springer-Verlag Berlin Heidelberg (1978).[17] S. P. Gristchacher. “The ring of coefficients for commutative complex K-theory”,arXiv:1611.03644, 2017.[18] W. M. Goldman “Topological components of spaces of representations”, Invent.Math. 93 (1988), 557–607.[19] H. Hironaka. “Triangulations of algebraic varieties”, Proc. Sympos. Pure Math. 29(1975) 165-185.[20] J. E. Humphreys. “Linear algebraic groups”, Graduate Texts in Mathematics, No. 21.Springer-Verlag, New York-Heidelberg, 1975.[21] T. Matumoto and M. Shiota. “Unique triangulation of the orbit space of a differentialtransformation group and its applications”, Adv. Stud. Pure Math. 9 (1986), 41-55.[22] J. P. May. “The geometry of iterated loop spaces”, Springer Lecture notes in mathe-matics, Springer, Berlin (1972).[23] J. P. May. “E∞-spaces, group completions and permutative categories”, New develop-ments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972) pages 61-93.London Math. Soc. Lecture Notes Ser., No11 Cambridge Univ. Press, London, 1974.[24] A. L. Onischnick and E. B. Vinberg. “Lie Groups and Algebraic Groups”, Translatedfrom the Russian by D. A. Leites, Springer-Verlag, 1990.[25] R. S. Palais. “The Classification of G-spaces”, Mem. Amer. Math. Soc. 36 (1960).[26] D.H. Park and D.Y. Suh. “Equivariant semi-algebraic triangulation of real algebraicG-varieties”, Kyushu J. Math. 50(1) (1996), 179-205.[27] D. Sjerve, E. Torres-Giese.“Fundamental groups of commuting elements in Liegroups”, Bull. London Math. Soc. (2008) 40 (1) 65-76.[28] B. Villarreal. “Cosimplicial groups and spaces of homomorphisms”, to appear inAlgebraic and Geometric Topology.72Appendix AAppendixA.1 Computing ker f in SINGULARHere we use the SINGULAR computer algebra system [15] to compute the kernelof the ring homomorphism used in the proof of Theorem 3.1.3:f : Z[c1,c2,y1,y2]→ Z[t,a1,a2,c,u]/(tu, ta1, ta2, tc,c2,u2,2u,cu,a1u)given by f (c1) = 2t +a1, f (c2) = t2+a2, f (y1) = 2c and f (y2) = a1c, as need inthe proof of 3.1.3.First we define the polynomial ringsP := Z[c1,c2,y1,y2] and F := Z[t,u,a1,a2,c],the ideal of F given by I := (tu, ta1, ta2, tc,c2,u2,2u,cu,a1u) and the quotient ringR := F/I.ring P = ZZ, (c1, c2, y1, y2), dp;ring F = ZZ, (t, u, a1, a2, c), dp;ideal I = t*u, t*a1, t*a2, t*c, cˆ2, uˆ2, 2*u, c*u,a1*u;qring R = std(I);In SINGULAR there is always a “current ring” and one defines homomorphisms73into it by specifying the domain and the list of images of the chosen generators forsaid domain:map f = P, 2*t+a1, tˆ2+a2, 2*c, a1*c;To ask for the kernel of the homomorphism, one sets the current to be thedomain of the homomorphism and uses the kernel function:setring P;kernel(R, f);SINGULAR responds with a list of generators for the ideal ker( f ):_[1]=y2ˆ2_[2]=y1*y2_[3]=y1ˆ2_[4]=c1*y1-2*y274

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