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Group actions on homotopy spheres Klaus, Michele 2011

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Group Actions on Homotopy Spheres by Michele Klaus B.S., Swiss Institute of Technology of Lausanne, 2005 M.S., Swiss Institute of Technology of Lausanne, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2011 c© Michele Klaus 2011 Abstract In the first part of this thesis we discuss the rank conjecture of Benson and Carlson (5.2 in [7]). In particular, we prove that if G is a finite p-group of rank 3 and with p odd, or if G is a central extension of abelian p-groups, then there is a free finite G-CW-complex X ' Sn1 × ... × Snrk(G) ; where rk(G) is the rank of G. We also treat an extension of the rank conjecture to groups of finite virtual cohomological dimension. In this context, for p a fixed odd prime, we show that there is an infinite group Γ satisfying the two following properties: every finite subgroup G < Γ is a p-group with rk(G) ≤ 2 and for every finite dimensional Γ-CW-complex X ' Sn there is at least one isotropy subgroup Γσ with rk(Γσ) = 2. In the second part of the thesis we discuss the study of homotopy G- spheres up to Borel equivalence. In particular, we provide a new approach to the construction of finite homotopy G-spheres up to Borel equivalence, and we apply it to give some new examples for groups of the form CpoCqr . ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The Rank Conjecture . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Equivariant obstruction theory . . . . . . . . . . . . . . . . . 7 2.2 A general construction . . . . . . . . . . . . . . . . . . . . . 13 2.3 Some p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Infinite groups . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Finite Homotopy G-Spheres up to Borel Equivalence . . . 34 3.1 Homological algebra over the orbit category . . . . . . . . . 34 3.2 Finite homotopy G-spheres . . . . . . . . . . . . . . . . . . . 39 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 iii Acknowledgements I wish to thank Professors Alejandro Adem and Jeff Smith for their sugges- tions, encouragement and patience. Thank you to Jim Clarkson and Professor Ian Hambleton for fruitful conversations about the topic of the second part of the thesis. iv Chapter 1 Introduction The origin of the study of group actions on spheres can be found in the spherical space form problem stated by H. Hopf in 1925. The problem con- cerns the classification of the finite groups that can act freely and smoothly on a sphere Sn. The first result was due to P.A. Smith [35], who showed that if a finite group G acts freely on a sphere, then it must have periodic cohomology. Another necessary condition was found by Milnor [31]: in such a group, every element of order two must be central. Later, Madsen, Thomas and Wall [30] proved that these two necessary conditions are also sufficient: a finite group G acts freely and smoothly on a sphere if and only if G has periodic cohomology and every involution is central. From the homotopy point of view, Swan [38], proved that a finite group G has periodic cohomology if and only if there is a free finite G-CW-complex X homotopy equivalent to Sn. These classical results have been extended in various directions. In this thesis we discuss and contribute to two such directions. In the first part we prove some results about the rank conjecture, while in the second part we discuss a new approach to the study of finite homotopy G-spheres 1 up to Borel equivalence. Despite dealing with similar objects, the two chapters are unrelated and can be read independently. We begin by introducing the topic of the first part. It is a classical result that a finite group has periodic cohomology if and only if all of its abelian 1A homotopy G-sphere is a G-CW-complex X homotopy equivalent to Sn. 1 Chapter 1. Introduction p-subgroups are cyclic. This last condition is commonly expressed using the notion of rank: The rank of a finite group G is the number rk(G) = max { k ∈ N| there is a prime p with (Z/p)k < G}. Thus, a finite group G has periodic cohomology if and only if rk(G) = 1. From the aforementioned result of Swan, it follows that a group G has rank one if and only if there is a free finite G-CW-complex X ' Sn 2. Based on some of their own algebraic results, Benson and Carlson [7] suggested a first extension of Swan’s theorem: the rank conjecture. The rank conjecture states that for any finite group G we have that rk(G) = hrk(G), where hrk(G) is the homotopy rank of G: min k∈N {there is a free finite dimensional G-CW-complex X ' Sn1 × ...× Snk} With this notation, Swan’s result says that rk(G) = 1 iff hrk(G) = 1. It is worth noting that all the ingredients in the definition of hrk(G) are essential for the conjecture to have a chance to be true. Every group G acts freely on some EG ' ∗ but EG has infinite dimension. The symmetric group Σ3 cannot act freely on any sphere Sn ([31]), but it does act freely on some finite X ' Sn ([38]), so that we require the G-CW-complex to only be homotopy equivalent to a product of spheres. Similarly the alternating group A4 cannot act freely on any finite dimensional X ' Sn×Sn ([32]), but it does act freely on some finite X ' Sn×Sm ([2]), so that we don’t require X to be homotopy equivalent to a product of equidimensional spheres. Some results are known about the rank conjecture: Heller [25] showed that (Z/p)3 cannot act freely on a finite dimensional CW-complex homotopy equivalent to a product of two spheres. Adem and Browder [5] showed that if (Z/p)m acts freely on (Sn)k, then m ≤ k. More recently, work of Adem and Smith [2] and Jackson [28], shows that if rk(G) = 2 then hrk(G) = 2 for a large family of groups. This includes the p-groups and the simple groups different from PSL3(Fp), p odd. In this context, our main theorems are the following: 2Throughout the thesis the symbol ”'” will mean ”homotopy equivalent”. 2 Chapter 1. Introduction Theorem 2.3.5. For p an odd prime, every finite p-group of rank three acts freely on a finite CW-complex homotopy equivalent to the product of three spheres. Theorem 2.3.8. Let G be a central extension of finite abelian p-groups. If rk(G) = r then hrk(G) ≤ r. Note that a converse to theorem 2.3.5 is given by Hanke in [23] in the sense that: if (Z/p)r acts freely on X = Sn1 × ...×Snk and if p > 3dim(X), then r ≤ k. Another way of extending Swan’s result is to consider infinite groups with periodic cohomology. In this case, results of Prassidis [33], Connolly and Prassidis [16] and Adem and Smith [2], show that a discrete group Γ acts freely and properly on Rn × Sm if and only if Γ is a countable group with periodic cohomology. Analogously, it is reasonable to ask for which other results (concerning the rank conjecture) can be extended to infinite groups. The best candidates to study, are groups Γ with finite virtual cohomo- logical dimension 3. The reason is that for every such group Γ, there is a contractible finite dimensional Γ-CW-complex EΓ, with finite isotropy sub- groups. The question for infinite groups seems more complicated than the one for finite groups for the following reason: every finite rank 2 p-group has a linear sphere with periodic isotropy subgroups. Our main result here states that, for infinite groups, the analogue property does not hold: Theorem 2.4.1. For p an odd prime, there is an infinite group Γ with finite virtual cohomological dimension, satisfying the two following proper- ties: every finite subgroup G < Γ is a p-group with rk(G) ≤ 2 and for every finite dimensional Γ-CW-complex X ' Sn there is at least one isotropy subgroup Γσ with rk(Γσ) = 2. 3We say that Γ has finite virtual cohomological dimension if there is Γ′ < Γ with |Γ/Γ′| finite and Hn(Γ′) = 0 for all coefficients and for all n big enough. 3 Chapter 1. Introduction We turn now our attention to the topic of the second part. Instead of requiring a free G-action and looking for a homotopy product of spheres that can sustain it; one can fix a homotopy sphere and ask for which dif- ferent G-actions can occur for a fixed group G. This has been done, for example, by tom Dieck and Petrie [42] who initiated the study of homotopy representations. Another example is given by Dotzel and Hamrick in [18], where they show that, for a p-group G, each finite dimensional homotopy G-sphere is equivalent, in some sense, to a linear one. In this setting, we need a way of comparing G-spaces: In general, for two G-spaces X and Y , to be equivariantly homotopy equivalent usually means the following: There are G-equivariant maps f : X → Y and g : Y → X and G-equivariant homotopies F : X × I → X from g ◦f to IdX and G : Y ×I → Y from f ◦g to IdY . Under this definition, the G-spaces EG and ∗ are not equivariantly homotopy equivalent because there is no equivariant map ∗ → EG. On the other hand, there is a G-equivariant map EG→ ∗ which is a homotopy equivalence. With this is mind, we recall the following definition: two G-spaces X and Y are Borel equivalent if the Borel constructions EG ×G X and EG ×G Y are weak equivalent over BG. One can show that this happens if and only if there is a zig-zag of G-maps X → Z1 ← ...→ Zk ← Y each of which is a homotopy equivalence. Clearly, in this setting, EG is equivalent to ∗. On the other hand, for any finite group G, Grodal and Smith [21] classi- fied all the possible homotopy G-spheres up to Borel equivalence. To state the classification theorems in [21], we need to quickly introduce the follow- ing notation. (More details are in section 3.1). For a group G, we let Γ be the orbit category OrG. This is the category with Ob(Γ) = {G/H|H < G} and Mor(G/K,G/H) = {f : G/K → G/H|f is a G-equivariant map}. For a prime p, the category Γp, is the full subcategory of Γ defined by: G/H ∈ Ob(Γp) if and only if H is a p-group. We also need to recall that to each homotopy G-sphere X, we can as- sociate a family of dimension functions { DimpX(−) } p||G| in the following 4 Chapter 1. Introduction way: for all p||G| and for all p-subgroups K, the homotopy fixed points XhK = MapK(EK,X) have the mod p homology of a sphere. This yields dimension functions DimpX(−) : Γp → N; G/K 7→ Dim(H∗(XhK ,Fp)) re- specting fusion and satisfying the Borel-Smith condition 4. With this nota- tions, the first classification theorem of [21] is: Theorem 1.0.1. [21] Let X and Y be two homotopy G-spheres. Assume that for all p||G| and for all p-subgroups K < G, we have that XhK and Y hK are connected. The space X is Borel equivalent to Y if and only if DimpX(−) = DimpY (−) for all p||G|. Moreover, every family of functions {Dp(−) : Γp → N}p||G| is realized as the dimension function family of a ho- motopy G-sphere, providing that: 1. Dp(G/e) = Dq(G/e) for all p, q||G|; 2. Dp(−) satifies the Borel Smith condition for all p||G|; 3. Dp(−) respects fusion for all p||G|. In the second classification theorem, we denote by C(Fp) the category of chain complexes of left F(p)-modules and we write C(FpΓp) for the category of contravariant functors F : Γopp → C(Fp). Theorem 1.0.2. [21] The family of functors {Φp}p||G| defined by Φp : {homotopy G-spheres} → C(FpΓp), with Φp(X)(G/K) = C∗(MapG(EG × G/K,X),Fp), satisfies the following properties: 1. For all X, the chain complex Φp(X) is quasi-isomorphic to a perfect FpΓp-chain complex. 2. Assume that for all p||G| and for all p-subgroups K < G, we have that XhK and Y hK are connected. The chain complexes Φp(Y ) and Φp(X) are quasi-isomorphic for all p||G|, if and only if X and Y are Borel equivalent. 4A function Dp(−) : Γp → N respects fusion, if Dp(K) = Dp(K′) whenever there is g ∈ G with gKg−1 = K′. We say that Dp satisfies the Borel-Smith condition if Dp(−)|P coincides with the di- mension function of an orthogonal P -representation for all p-Sylow P . 5 Chapter 1. Introduction Because of these theorems, it seems now possible to develop a finite- ness obstruction theory for homotopy G-spheres up to Borel equivalence, in the realm of homological algebra over the orbit category. Such a prob- lem has already been attacked by Jim Clarkson [14]. In particular, he was able to prove that all homotopy G-spheres are finite dimensional, up to Borel equivalence. Moreover, if G = Cp o Cq, he also showed that a homo- topy G-sphere X is finite, up to Borel equivalence, if and only if 2q divides DimpX(G/e)−DimpX(G/Cp). His method involves Dold’s theory of algebraic Postnikov towers and relies on the assumption that |G| = pq. Inspired by Clarkson’s work [14], and using some of his results, we sug- gest a strategy where Postnikov towers are replaced by an arithmetic square. In particular, we provide a new approach to the construction of finite ho- motopy G-spheres, and we apply it to give new examples for groups of the form Cp o Cqr . Theorem 3.2.6. Consider the group G = CpoCqr with faithful Cqr action on Cp. For all s ≤ r and for all j ≥ 3, there is a finite homotopy G-sphere X with: DimqX(G/Cqt) = { j + 2qr if t ≤ s, j otherwise. while DimpX(G/Cp) = j. 6 Chapter 2 The Rank Conjecture In the first section we introduce some background on equivariant obstruction theory. In the second section we use equivariant obstruction theory to prove the following auxiliary result: Let p be an odd prime, let G be a p-group and S(V ) a complex representation G-sphere. Then, for all integers k ≥ 0, there exists a positive integer q such that the group pik(AutG(S(V ⊕q))) is finite. We then incorporate this result in an outline of a known construction ([2], [16], [43]) that, in favourable conditions, gives a strategy to build group actions on products of spheres with controlled isotropy subgroups. In the third section we use this construction twice: once to prove theorem 2.3.5 and once to generalize theorem 3.2 in [2] for p-groups G: if X is a finite dimensional G-CW-complex with abelian isotropy, we show that there is a free finite dimensional G-CW-complex Y ' X×S1×...×Snk . As a corollary we will be able to prove theorem 2.3.8. Finally, in the fourth section, we discuss the extension of the rank con- jecture to infinite groups and we prove theorem 2.4.1. 2.1 Equivariant obstruction theory We now introduce some notions and results of equivariant obstruction the- ory that will be used in the sequel. In our outline we follow the classical references [40] and [41]. We include this section in order to make the proof of proposition 2.2.5 more readable. To this end, we are not going to state all the results in full generality. 7 2.1. Equivariant obstruction theory Throughout this section, G will denote a finite group. A G-CW-complex is a CW-complex X together with a G-action such that: 1. For each g ∈ G and each open cell E of X, the left translation gE is again an open cell of X, 2. If gE = E, then the induced map E → E, x 7→ gx is the identity. There areG-actions on CW-complexes which satisfy (a) but not (b). Usually, for a suitable subdivision, (b) is then satisfied. A pair of G-CW-complexes is a pair of CW-complexes (X,A) for which A and X are G-CW-complexes and the inclusion A → X is G-equivariant. The r-th skeleton of a pair (X,A) is the space Skr(X,A) = Skr(X) ∪ A. To shorten the notation, we will write Xr = Skr(X,A). Finally, we write Dim(X,A) for the biggest dimension among the cells of X not in A. The main object of study here is the homology and cohomology of such a pair (X,A). As usual, there is the cellular definition, suitable for defining the groups of the chain and cochain complexes, and there is the singular definition, suitable for the description of the differentials in the chain and cochain complexes. The two approaches agree, but we will not enter in the details of why they do so. We begin with the more formal singular definition. Let (X,A) be a pair of G-CW-complexes with free G-action on X \A. As usual, see for instance [24], intertwining the exact sequences of singular homology groups for the CW-pairs (Xn, Xn−1), we recover a chain complex of ZG modules C∗(X,A): ... // Hn+1(Xn+1, Xn) // Hn(Xn, Xn−1) // ... If M is another ZG-module, the cochain complex: C∗G(X,A;M) = HomZG(C∗(X,A),M) yields cohomology groups H∗G(X,A;M). 8 2.1. Equivariant obstruction theory We describe now the groups Cn(X,A) in cellular terms. Since we as- sumed that the G-action on X \ A is free, Xn is obtained from Xn−1 as a pushout: ∐ i∈J G× Sn−1 //  Xn−1 ∐ i∈J G×Dn // Xn The corresponding characteristic map: φ : ∐ i∈J G× (Dn, Sn−1)→ (Xn, Xn−1) provides a canonical basis for the free ZG-module Hn(Xn, Xn−1). It is given by the images of the canonical generators of Hn(Dn, Sn−1) under the j-th component of φ: {e} × (Dn, Sn−1) // G× (Dn, Sn−1) φj // (Xn, Xn−1) . An element of CnG(X,A;M) may thus be identified with a function on this basis with value in M . Remark 2.1.1. It is worth noting that the ZG-module M can be thought of as a local coefficients system over X/G\A/G. Consequently, HnG(X,A;M) ∼= Hn(X/G,A/G;M) (see section 8 of chapter 2 in [41]). We have so far talked about generic coefficients M . For our goals, the module M can be supposed to be of the following form: Let Y be a G-CW- complex. If Y is path-connected and n-simple, i.e. pi1(Y, y) acts trivially on pin(Y, y), then the canonical map pin(Y, y) → [Sn, Y ] from pointed to free homotopy classes is bijective. The action of G on Y induces therefore a well defined action of G on pin(Y ) and we can consider cochain complexes of the form C∗G(X,A, pinY ). Obstruction classes live in such a complex. We will be interested in studying obstructions to extending homotopies. To this end, 9 2.1. Equivariant obstruction theory we first need to talk about obstructions to extending maps. The main result here is theorem 3.10 of [41]: Theorem 2.1.2. Let Y be a simply connected G-CW-complex. Let (X,A) be a path-connected pair of G-CW-complexes with G acting freely on X \A. For each n ≥ 1 there exists an exact obstruction sequence: [Xn+1, Y ]G → Im([Xn, Y ]G → [Xn−1, Y ]G) c n+1−−−→ Hn+1G (X,A;pinY ) which is natural in (X,A) and Y . The exactness of this sequence means that each homotopy class Xn−1 → Y , which is extendable over Xn, has an associated obstruction element in the group Hn+1G (X,A;pinY ). This obstruction element is zero if and only if the homotopy class Xn−1 → Y is extendable over Xn+1. Proof. Details can be found in [41]. Here we outline the construction of the map cn+1, because it will be relevant to define the obstruction classes that we are interested in. Let φ : ∐ i∈J G × (Dn+1, S1) → (Xn+1, Xn) be the characteristic map and ej ∈ C∗(X,A) be the basis element corresponding to the j-th component φj . Fix an element h ∈ [Xn, Y ]G. The composition h◦φj defines an element cn+1(h)(ej) ∈ [Sn, Y ] = pinY . Extending by linearity, we recover an element cn+1(h) ∈ Cn+1G (X,A;pinY ). This is not enough to define the announced map cn+1, We also need to show that, if [h0] and [h1] are elements of [Xn, Y ] with the same image in [Xn−1, Y ], then cn+1(h0) and cn+1(h1) differ by a coboundary. For that purpose, choose a G-homotopy k : I×Xn−1 with ki = hi|Xn−1 . Suppose that ϕ : (Dn, Sn−1)→ (X,Xn−1) is the characteristic map of an n-cell defining a basis element e ∈ Cn(X,A). By composition, we recover a map: {0} ×Dn ∪ I × Sn−1 ∪ {1} ×Dn ϕ×Id // {0} ×Xn ∪ I ×Xn−1 ∪ {1} ×Xn (h0,k,h1)  Y 10 2.1. Equivariant obstruction theory Composing with the standard homeomorphism Sn ∼= ∂I ×Dn ∪ I × Sn−1, yields a homotopy class x ∈ [Sn, Y ] = pinY . Setting d(h0, k, h1) : Cn(X,A)→ pinY ; e 7→ x we recover an element d(h0, k, h1) ∈ CnG(X,A;pinY ). It turns out that δd(h0, k, h1) = cn+1(h0)− cn+1(h1) as required. As a by-product, the end of the proof also provides an obstruction cochain d(h0, k, h1) ∈ CnG(X,A;pinY ), defined for two maps h0, h1 : X → Y and a G-homotopy k : I×Xn−1 → Xn−1, with ki = hi|Xn−1 . The properties of such an obstruction cochain are given in [41] and are summarized by: Proposition 2.1.3. With the notation above, the cochain d(h0, k, h1) is a cocyle with homology class d̄(h0, k, h1) ∈ HnG(X,A;pinY ). Moreover: 1. d̄(h0, k, h1) = −d̄(h1, k−, h0), where k− is the inverse homotopy, 2. d̄(h0, k, h1) + d̄(h1, k′, h2) = d̄(h0, k + k′, h2), 3. d̄(h0, k, h1) = 0 if and only if the G homotopy k extends to a G- homotopy K : I ×Xn → Xn with Ki = hi|Xn. We have being working under the assumption the G acts freely on X \A, so far. Following [40] we are going to explain now why this is not a ma- jor restriction. As usual, G is a finite group, X a G-CW-complex and Y a simply connected G-CW-complex. Choose an indexing of the con- jugacy classes of isotropy subgroups {(H1), ..., (Hm)} such that if (Hj) < (Hi) then i < j. Consider the filtration X1 ⊂ ... ⊂ Xm given by Xi = {x ∈ X| (Gx) = (Hj) for some j ≤ i}. Such a filtration allows us to re- cover pairs (XHii , X H1 i−1) with free WHi = NHi/Hi-action on X Hi i \ XHii−1. Moreover: Proposition 2.1.4. [40] The WHi = NHi/Hi-action on XHii \XHii−1 is free. Furthermore, given a G-map k : Xi−1 → Yi−1, the extensions K : Xi → Yi of f , are in bijective correspondence with the WHi-extension e : XHii → Y Hii of k : XHii−1 → Y Hii−1. 11 2.1. Equivariant obstruction theory Proof. Set XHi = {x ∈ X| (Gx) = (Hi)}. Given K, we have e = KHi and since GXHi = Xi \ Xi−1, the G map K is uniquely determined by KHi . Which shows injectivity. Conversely, suppose that we are given a WHi-map e : XHii → Y Hii extending kH . We define a map E : Xi → Yi by: x 7→ { e(x), if x ∈ Xi−1; ge(y), if x = gy with y ∈ XHi . Following proposition 8.1.5 of [40], one can show that E is well defined and continuous. We end the section by summarizing its application to the proof of 2.2.5: Consider two G-maps f1, f2 : X → Y . Assume that there is i0 such that f1|Xi0 : Xi0 → Yi0 and f2|Xi0 : Xi0 → Yi0 are homotopic. In order to know if f1 and f2 are homotopic, we want to know if the successive restrictions f1|Xi : Xi → Yi and f2|Xi : Xi → Yi are homotopic for all i > i0. Since homotopies are G-maps F : I × X → Y with trivial G-action on I, we can consider (I × X)i = I × Xi. A G-extension K : I × Xi → Yi of the given G-homotopy k : I ×Xi−1 → Yi−1, exists if and only if there is a WHi-extension of kHi : I ×XHii−1 → Y Hii−1, by proposition 2.1.4. By theorem 2.1.3, such a WHi-extension exists, if and only if the dif- ference cocycles d̄((f1)i, kHi , (f2)i) ∈ HnG(XHii , XHii−1;pinYi) are all zero, for n = 1, ..., Dim(XHii , X Hi i−1). By virtue of remark 2.1.1, this homology groups satisfy: HnG(X Hi i , X Hi i−1;pinYi) ∼= Hn(XHii /WHi, XHii−1/WHi;pinYi), where pinYi is interpreted as a local coefficients system over XHii \XHii−1. In our case, the space Y will be a complex linear sphere, so that the local coefficients above are actually untwisted. For further reference, notice also that XHii−1 = ∪H>HiXH while XHii = XHi . 12 2.2. A general construction 2.2 A general construction The main result of this section is the construction of proposition 2.2.6. A key ingredient of the construction is proposition 2.2.5, which says that under some conditions pik(AutG(Sn)) is finite. We begin with some quick gener- alities. For a G-space X we write AutG(X) for the monoid of equivariant self-homotopy equivalences of X. In other words, an element of AutG(X) is an equivariant map f : X → X which is an homotopy equivalence and for which there exists an homotopy inverse g : X → X and homotopies F,H : X × I → X, between f ◦ g and IdX , and between IdX and g ◦ f , which are all equivariant. We are interested in the monoid AutG(X) because, for a space Y , we have an injection of the G-equivariant X-fibrations over Y into [Y,BAutG(X)] (see [6]). In particular, in the construction of proposition 2.2.6, we will need to extend equivariantly a spherical fibration from Sn−1 to Dn. Thus we want to study the groups [ Sn−1, BAutG(Sm) ] = pin−1(BAutG(Sm)) = pin−2(AutG(Sm)) The sequel of this section is structured with a series of lemmas and corollaries that we assemble into a proof of proposition 2.2.5. Lemmas 2.2.1, 2.2.2 and 2.2.4 are individual results needed in the proof of proposition 2.2.5. Lemma 2.2.3 serves the proof of lemma 2.2.4. Lemma 2.2.1. Let X be a G-CW-complex and let AutG(X) be the monoid of G-equivariant self-homotopy equivalences of X. For each f ∈ AutG(X) we write AutG(X)f for the path component of f . For k > 0, the map of unbased homotopy classes ϕ : [ Sk, AutG(X)f ] → [Sk ×X,X] G is injective and factors through: [ Sk, AutG(X)f ]  ϕ // [ Sk ×X,X] G pik(AutG(X)f ) 66mmmmmmmmmmmmm 13 2.2. A general construction In particular all G-equivariant homotopies H : I×Sk×X → X between maps representing the same element in Im(ϕ) can be taken to satisfy H(t, ∗, x) = H(t′, ∗, x) for all t, t′ ∈ I and x ∈ X. Proof. The map ϕ : [ Sk, AutG(X)f ] → [Sk ×X,X] G is clearly well de- fined. To see that it is injective, consider a G-equivariant homotopy H : I×Sk×X → X from ϕ(g1) to ϕ(g2). ClearlyH|{0}×{x0}×X = ϕ(g1)(x0,−) = g1(x0) ∈ AutG(X)f . Which implies that H|{t}×{x}×X ∈ AutG(X)f for all (t, x) ∈ I × Sk because H|{t}×{x}×X ' H|{0}×{x0}×X via a path in I × Sk from (0, x0) to (t, x). As a result, H defines an homotopy from g1 to g2. To prove that ϕ factors through: [ Sk, AutG(X)f ]  ϕ // [ Sk ×X,X] G pik(AutG(X)f ) 66mmmmmmmmmmmmm we want to show that the map pik(AutG(X)f ) → [ Sk, AutG(X)f ] is a bijection. Observe that AutG(X) is a monoid, thus an H-space so that pi1(AutG(X)Id) acts trivially on pik(AutG(X)Id). The monoid AutG(X) is very nice because all of its connected components are homotopy equivalent through maps of the form: AutG(X)Id → AutG(X)f with g 7→ f ◦g. Conse- quently pi1(AutG(X)f ) acts trivially on pik(AutG(X)f ) for all f ∈ AutG(X). We conclude that pik(AutG(X)f )→ [ Sk, AutG(X)f ] is a bijection. The last claim directly follows from the diagram. Lemma 2.2.2. Let G be a finite group acting on a space X. Let H1 < G be an isotropy subgroup maximal among isotropy subgroups. Set X1 = {x ∈ X| Gx ∈ (H1)}, where (H1) denotes the conjugacy class of H1. We then have that AutG(X1) ∼= AutWH1(XH1) (here WH1 = NH1/H1 is the Weil group). Proof. Let’s begin by studying X1. Clearly X1 ⊂ ∪H∈(H1)XH . Since H ∈ (H1) is supposed to be maximal, we must have that if x ∈ XH , then Gx = H 14 2.2. A general construction so that X1 = ∪H∈(H1)XH . Similarly, if x ∈ XH ∩XH ′ , for H, H ′ in (H1), then H = Gx = H ′. As a result X1 = ∪H∈(H1)XH . Observe next that a G-equivariant map f : X1 → X1 restricts to aWH1- equivariant map f1 : XH1 → XH1 because WH1 = NH1/H1 and H1 acts trivially on X1. The same holds for a G-equivariant homotopy F : I×X1 → X1, so that we have a well defined map res : AutG(X1)→ AutWH1(XH1). One can then show that the map res : AutG(X1) → AutWH1(XH1) has an inverse given by res−1(f)(x) = gf(g−1x), where g ∈ G is such that g−1x ∈ XH1 . Lemma 2.2.3. Let G be a finite group and Sn a linear G-sphere. If 0 < k < n then Hn(Sk × Sn/G, {∗} × Sn/G,Z) is finite. Proof. Consider the long exact sequence of the pair (Sk×Sn/G, {∗}×Sn/G) with integer coefficients: Hn−1(Sk × Sn/G) // // Hn−1({∗} × Sn/G)  Hn(Sk × Sn/G) i∗  Hn(Sk × Sn/G, {∗} × Sn/G)oo Hn({∗} × Sn/G) Clearly Hn(Sk × Sn/G, {∗} × Sn/G) ⊂ Ker(i∗). But Hn(Sk × Sn/G) ∼= Hn({∗} × Sn/G) ⊕ Hn−k({∗} × Sn/G). Thus for i∗ : Hn(Sk × Sn/G) → Hn({∗} × Sn/G) we have that Ker(i∗) ∼= Hn−k({∗} × Sn/G). Finally, the groups Hn−k(Sn/G) are finite for 0 < k < n because Hn−k(Sn/G,Q) = 0 by the Vietoris-Begle theorem. Lemma 2.2.4. Let G be a finite group and S(V ) a linear G-sphere. For H ≤ G write nr(H) for the integer such that S(V ⊕r)H = Snr(H). For all 15 2.2. A general construction k > 0 there is an integer q > 0 such that the groups: Hnq(Hi)(Sk×Snq(Hi)/WHi,∪H>HiSk×Snq(H)/WHi∪{?}×Snq(Hi)/WHi,Z) are finite for all Hi with n1(Hi) > 0. Proof. Fix a subgroup Hi < G such that n1(Hi) > 0. If there is H > Hi with n1(H) = n1(Hi), then the required cohomology group is zero (it is of the form Hn(Hi)(X,X,Z)). Assume that for all H > Hi we have n1(H) < n1(Hi). In this case we want so show that we can take enough direct sums to be in the situation of lemma 2.2.3. Let nr,i = maxH>Hi {nr(H)} and mr,i = nr(Hi) − nr,i > 0. Observe that nr(H) = rn1(H) + (r − 1) so that nr,i = rn1,i + (r − 1) and mr,i = nr(Hi)−nr,i = rn1(Hi)+(r−1)−(rn1,i+(r−1)) = rm1,i. Therefore there is a qi big enough such that mr,i > k+2. In other words nr(Hi)−k−2 > nr,i. We have found an integer qi > 0 such that all the cells τ of the CW-complex Snqi (Hi) of dimension dim(τ) ≥ nqi(Hi)− k− 2, are also cells of the relative CW-complex (Snqi (Hi),∪H>HiSnqi (H)). We turn now our attention to the announced cohomology group. By our condition on the cells of Snqi (Hi), we have that the cells τ of the CW- complex Sk×Snqi (Hi)/WHi of dimension dim(τ) ≥ nqi(Hi)−2, are also cells of the relative CW-complex (Sk×Snqi (Hi)/WHi,∪H>HiSk×Snqi (H)/WHi). Henceforth: Hnqi (Hi)(Sk ×Snqi (Hi)/WHi,∪H>HiSk ×Snqi (H)/WHi ∪ {?}× Snqi (Hi),Z) = Hn(Hi)(Sk×Snq1 (Hi)/WHi, {?}×Snqi (Hi)/WHi,Z). This last group is finite, by virtue of lemma 2.2.3. We conclude by observing that we can then set q = maxHi<G {qi}. Proposition 2.2.5. Let G be a finite p-group. Let S(V ) be a complex representation G-sphere. For all integers k ≥ 0 there exists an integer q > 0 such that pik(AutG(S(V ⊕q))) is finite. Proof. If k = 0, the result has been proven in [19]. Assume that k > 0. Before explaining how the proof proceeds, we recall some notation: Choose 16 2.2. A general construction an ordering of the conjugacy classes of isotropy subgroups {(H1), ..., (Hm)} such that if (Hj) < (Hi) then i < j. Consider the filtration S(V )1 ⊂ ... ⊂ S(V )m = S(V ) given by S(V )i = {x ∈ S(V )| (Gx) = (Hj); j ≤ i}. We have homomorphisms Ri : pik(AutG(S(V ))) → pik(AutG(S(V )i)) and Si : pik(AutG(S(V )i)) → pik(AutG(S(V )i−1)). Here is how the proof runs. Look at the commutative diagram: pik(AutG(S(V ))) Id // Ri %%KK KK KK KK KK KK KK KK KK KK K R1 8 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 pik(AutG(S(V )m)) ... Si+1  pik(AutG(S(V )i)) Si ... S2  pik(AutG(S(V )1)) Clearly to prove that pik(AutG(S(V ))) is finite is the same as to prove that Im(Rm) is finite. To prove that Im(Rm) is finite, we will show by induction over i that Im(Ri) is finite. Such an induction can be performed by showing that Im(R1) is finite and that S−1i (Ri−1(f)) ∩ Im(Ri) is finite for all i and for all f ∈ pik(AutG(S(V ))). This outline can only be carried out up to replacing S(V ) with some power S(V ⊕q). We begin by showing that there is q1 > 0 such that pik(AutG(S(V ⊕q1)1)) is finite. In particular we will have that Im(R1) ⊂ pik(AutG(S(V ⊕q1)1)) is finite. For H < G write nr(H) for the integer such that S(V ⊕r)H = Snr(H). Observe that nr(H) = rn(H) + (r − 1). By lemma 2.2.2 we have that pik(AutG(S(V )1)) = pik(AutWH1(S n1(H1))). The WH1-action on Sn1(H1) is free because H1 is maximal among isotropy subgroups. Therefore proposi- tion 2.4 of [16] says that pik(AutWH1(S n1(H1))) is finite if k < n1(H1)− 1. If k ≥ n1(H1)−1, then there is a q1 > 0 for which k < q1n1(H1)+(q1−1)−1 = nq1(H1) − 1. As a result pik(AutG(S(V ⊕q1)1)) = pik(AutWH1(Snq1 (H1))) is 17 2.2. A general construction finite (always by proposition 2.4 of [16]). Assume that we showed that Im(Ri−1) is finite. The inductive step is to prove that there is q ≥ qi−1 such that S−1i (Ri−1(f)) ∩ Im(Ri) is finite for all i and for all f ∈ pik(AutG(S(V ⊕q))). For that purpose we are going to use equivariant obstruction theory a la Tom Dieck (see [40] section 8 and [41] chapter 2). We begin with some preliminaries. As in lemma 2.2.4, let q′ > 0 be such that the groups: Hnq′ (H ′)(Sk×Snq′ (H′)/WH ′,∪H>H′Sk×Snq′ (H)/WH ′∪{?}×Snq′ (H′)/WH ′) are finite for all H ′ < G with n1(H ′) > 0. Let q = max {q1, q′}. To simplify the notation we write W = V ⊕q, X = Sk × S(W ) and X̄Hi = ∪H>HiXH ∪ {?} × S(W )Hi . With this notation we have that the group: Hnq(Hi)(XHi/WHi, X̄Hi/WHi, pinq(Hi)(S nq(Hi))) is finite by lemma 2.2.4, while if r 6= nq(Hi) then the groups: Hr(XHi/WHi, X̄Hi/WHi, pir(Snq(Hi))) are finite because they are finitely generated torsion abelian groups. (The fixed points of a complex representation spheres are odd-dimensional spheres whose homotopy groups are all but one finite). A word of explanation is in order here: the space XHi is the one over which we want to extend a map already defined on X̄Hi (see end of section 2.1). We have a map over the part of X̄Hi given by ∪H>HiXH , because of the inductive hypothesis. We have a map over the part of X̄Hi given by {∗} × S(W )Hi , because all the maps come from pik(AutG(S(W ))) in the following way: By lemma 2.2.1 there is an injection pik(AutG(S(W )i)) → 18 2.2. A general construction [ Sk × S(W )i, S(W )i ] G yielding a diagram with injective columns: pik(AutG(S(W )i)) Si // ϕi  pik(AutG(S(W )i−1)) ϕi−1  [Xi, S(W )i]G si // [Xi−1, S(W )i−1]G To prove that S−1i (Ri−1(f)) ∩ Im(Ri) is finite, it is enough to prove that s−1i (ϕi−1(Ri−1(f)))∩ϕi(Im(Ri)) is finite. By abuse of notation we will keep on writing Si and Ri−1(f), but we will think of them as living in the bottom row of the diagram. Now, a homotopy h between Ri−1(f) and Ri−1(g) is con- stant over {∗}×S(W )i−1 where it coincides with both Ri−1(f) and Ri−1(g). Consequently, h can be extended to a homotopy from Ri(f)|Xi−1∪{∗}×S(W )i to Ri(g)|Xi−1∪{∗}×S(W )i . Henceforth, writing Xi−1 ∪S(W )i for Xi−1 ∪{∗}×S(W )i, we can apply equivariant obstruction theory inductively over r to each of the diagrams: [Skr+1(Xi, Xi−1 ∪ S(W )i), S(W )i]G Si,r+1 // Skr  [Xi−1, S(W )i−1]G [Skr(Xi, Xi−1 ∪ S(W )i), S(W )i]G Si,r 33fffffffffffffffffffffff If r = 0, then Sk0(Xi, Xi−1 ∪ S(W )i) = Xi−1 ∐ {x0, ..., xl}. Consequently, S−1i,0 (Ri−1(f)) ∩ Sk0(Im(Ri)) depends on the connected components of the space S(W )i−1. But S(W )i−1 has finitely many connected components because it is a finite CW-complex, therefore S−1i,0 (Ri−1(f)) ∩ Sk0(Im(Ri)) is finite. From now on, to simplify the notation, we are going to write fi = Ri(f) for all possible i and f . Assume that S−1i,r (fi−1)∩Skr(Im(Ri)) ={ g1i,r, ..., g t i,r } is finite of order t (i.e. gji,r 6= gli,r if j 6= l). For each gi,r+1 ∈ S−1i,r+1(fi−1) ∩ Skr+1(Im(Ri)) there is a unique gji,r and a homo- topy h from gi,r = gi,r+1|Skr(Xi,Xi−1∪S(W )i) to gji,r Notice that, by definition, we have gj , g ∈ pik(AutG(S(W ))) with gi,r+1 = gi|Skr+1(Xi,Xi−1∪S(W )i) and gji,r = g j i |Skr(Xi,Xi−1∪S(W )i). 19 2.2. A general construction We write d̄(gi,r, h, g j i,r) ∈ Hr+1(XHi/WHi, X̄Hi/WHi, pir+1((Sn)Hi)) for the homology class of the difference cocyle as in the preceding section. Now, if d̄(g′i,r+1, h ′, Skr+1(g j i )) = d̄(gi,r+1, h, Skr+1(g j i )), then we have: d̄(g′i,r+1, h ′+h−1, gi,r+1) = d̄(g′i,r+1, h ′, Skr+1(g j i ))+ d̄(Skr+1(g j i ), h −1, gi,r+1) = d̄(g′i,r+1, h ′, Skr+1(g j i ))− d̄(gi,r+1, h, Skr+1(gji )) = 0 so that gi,r+1 ' g′i,r+1. We can therefore define an injection: S−1i,r+1(fi−1) ∩ Skr+1(Im(Ri)) ∐t j=1 { (gji,r) } ×Hr+1(XHi/WHi, X̄Hi/WHi, pir+1(Snq(Hi))) by setting gi 7→ { gji } × d̄(gi,r+1, h, Skr+1(gji )). Since we chose the integer q in order to have all the cohomology groups on the right hand side to be finite, we must have that the left hand side is finite as well. Summarizing, by induction we have that S−1i,r+1(fi−1) ∩ Skr+1(Im(Ri)) is finite for all r. Since X is finite dimensional, this shows that S−1i (fi−1)∩ Im(Ri) is finite. We conclude as explained in the outline at the beginning of this proof. We can now turn our attention to the main result of this section, the construction of proposition 2.2.6. We begin by giving a brief summary of the goal of the construction: We give conditions under which it is possible to ”attach” a linear sphere to a finite dimensional G-complex X, in such a way that the final result is a finite dimensional G-CW-complex Y ' X × Sn, whose isotropy groups are smaller than the one of the original space X. To keep track of the evolution of the isotropy subgroups in the process of attaching spheres, we introduce the following notation: 20 2.2. A general construction Let G be a finite group and X a G-CW-complex. We write: rkX(G) = max {n ∈ N| there exists Gσ with rk(Gσ) = n} Proposition 2.2.6. Let G be a finite p-group and let X be a finite dimen- sional G-CW-complex. Assume that to each isotropy subgroup Gσ we can associate a representation ρσ : Gσ → U(n) such that ρσ|Gτ ∼= ρτ when- ever Gτ < Gσ. If ρσ is fixed point free for all Gσ with rk(Gσ) = rkX(G), then there exists a finite dimensional G-CW-complex E ∼= X × Sm with rkE(G) = rkX(G)− 1. Moreover, if X is finite then E is finite as well. Proof. The proof follows [16]. We refer the reader to [43] for the details. Write S2n−1σ for the linear sphere associated to ρσ. We want to glue these spheres into a G-equivariant spherical fibration over X. We will proceed by induction over the skeleton of X. For every G-orbit of the 0-skeleton, choose a representative σ and define a mapG×GσS2n−1σ → X0 by (g, x) 7→ g·σ. This defines a G-equivariant spherical fibration S2n−1 → E0 → Sk0(X) whose total space is a finite dimensional G-CW-complex. Clearly if ρσ is fixed point free for all Gσ with rk(Gσ) = rkX(G), then rkE0(G) = rkX(G)− 1. The inductive step is next. Suppose given a G-equivariant spherical fibration over the (k − 1)-skeleton ∗qk−1S2n−1 → Ek−1 → Skk−1(X) whose total space is a finite dimensional G-CW-complex. Assume also that if ρσ is fixed point free for all Gσ with rk(Gσ) = rkX(G), then rkEk−1(G) = rkX(G) − 1. Now, for every G-orbit of a k-cell, choose a representative σ. The Gσ-equivariant fibration ∗qk−1S2n−1 → Ek−1|∂σ → ∂σ is classified by an element aσ ∈ pik−2(AutGσ(∗qk−1S2n−1)). We want to have aσ = 0: Observe that, in general, for two complex G-spheres S(V ) and S(W ), we have that S(V ⊕ W ) ∼= S(V ) ∗ S(W ) as G-spheres. Therefore, by proposition 2.2.5, we can take enough Whitney sums of the fibration ∗qk−1S2n−1 → Ek−1 → Skk−1(X) to guarantee that aσ = 0 (see lemma 2.3 and proposition 2.4 in [16]). We can then extend the Gσ-equivariant fibration ∗qkS2n−1 → Ek−1|∂σ → ∂σ equivariantly across the cell σ. We define a G-equivariant spherical fibration over the orbit of σ by 21 2.2. A general construction G×Gσ ∗qkS2n−1σ → Gσ with (g, x) 7→ g · σ. Repeating the procedure for all the representatives of the G-orbits of k-cells, we recover a G-equivariant spherical fibration ∗qkS2n−1 → Ek → Skk(X) with total space a finite dimensional G-CW-complex. Clearly if ρσ is fixed point free for all Gσ with rk(Gσ) = rkX(G), then rkEk(G) = rkX(G)−1. We conclude noticing that, by proposition 2.8 in [2], up to taking further fiber joins, we can assume that the total fibration ∗qS2n−1 → E → X is a product one. For the last statement, one can observe that all the constructions take place in the category of finite CW-complexes, providing that the initial space X is a finite CW-complex. We end the section by connecting the construction to our problem. Ob- serve first that rkX(G) = 0 if and only if the action is free. Assume that X ' Sn1 × ... × Snt is a finitely dimensional G-CW-complex with rkX(G) = rk(G) − t. In this case we want to show that the conditions of proposition 2.2.6 are fulfilled in order to recover a suitable space Y1 ' Sn1 × ... × Snt × Snt+1 with rkY1(G) = rk(G) − (t + 1). Clearly, we then wish to apply proposition 2.2.6 again and again until we recover a suitable free space Yrk(G)−t ' Sn1 × ...× Snt × Snrk(G) . In order to have the required G-space X to begin our process with, we consider the center Z(G) of G. Since G is a p-group, Z(G) is not trivial and acts freely on some Sn1 × ... × Snt where each Sni is the linear sphere of a representation ρi of Z(G). Consider the induced representations ηi = IndGZ(G)ρi. We recover a G-space X = S m1 × ... × Smt , where Smi is the linear G-sphere corresponding to ηi and rkX(G) = rk(G)− t. The question of deciding which p-groups satisfy the additional conditions given by repeatedly applying proposition 2.2.6, seems considerably harder. In the next chapter we exhibit two families of p-groups for which the condi- tions are satisfied. 22 2.3. Some p-groups 2.3 Some p-groups Before applying the strategy outlined at the end of the previous section to odd order rank three p-groups, we need some group theory. It comes from an unpublished work of Jackson [28] and we reproduce it here: Lemma 2.3.1. If G is a finite p-group with rk(G) = 3 and rk(Z(G)) = 1, then there exists a normal abelian subgroup Q < G of type (p, p) with Q ∩ Z(G) 6= 0. Proof. This follows from statements 4.3 and 4.5 in Suzuki [37]. Proposition 2.3.2. Let G be a finite p-group with p > 2, rk(G) = 3 and rk(Z(G)) = 1. Let Q be an abelian normal subgroup of type (p, p) as above. Suppose that H < G with H ∩ Z(G) = 0 and |H| = pn. Then either H is cyclic, H < CG(Q), H is abelian of type (p, pn−1) or H ∼= M(pn) =< x, y|xpn−1 = yp = 1, y−1xy = x1+pn−2 >. Proof. If rk(H) = 1 then H is cyclic since p > 2. Suppose that rk(H) = 2 and H ∩ Q 6= 0. By assumption Z(G) ∩ Q = Z/p, H ∩ Z(G) = 0 and Q ∩ H = Z/p. The map c : H → Aut(H ∩ Q) given by ch(x) = hxh−1 is well defined because Q is normal. Since |H| = pn and |Aut(H ∩Q)| = p− 1, we have that the map c is trivial. As a result we have that H < CG(Q). Assume now thatH∩Q = 0. In this caseH∩CG(Q) 6= H since otherwise we would have rk(G) > 3. Set K = H ∩CG(Q) and observe that K is cyclic (else we would have rk(G) > 3). Assume for a moment that [G : CG(Q)] = p. In this case [H : K] = p, in other words H has a maximal cyclic subgroup. By [37] section 4, H needs then to be abelian of type (p, pn−1) or M(pn). We still have to prove that [G : CG(Q)] = p. The group G acts on Q by conjugation and for each element q of Q not in center of G, we have that Gq = CG(Q). As a result |CG(Q)| = |Gq| = |G|/p since Q ∼= (Z/p)2 with the first coordinate in the center Z(G). 23 2.3. Some p-groups Proposition 2.3.3. Let G be a finite p-group with p > 2, rk(G) = 3 and rk(Z(G)) = 1. There exists a class function β : G → C such that for any subgroup H ⊂ G, with H ∩ Z(G) = 0, the restriction β|H is a complex character of H. If in addition H is a rank two elementary abelian subgroup, then the character β|H corresponds to an isomorphism class of fixed-point free representations. Proof. Define β : G→ C as follows: x 7→  (p2 − p)|G|, if x = 0; 0, if x ∈ Z(G) \ 0; −p|G|, if x ∈ Q \ Z(G); 0, if x ∈ CG(Q) \Q; −|G|, if x ∈ G \ CG(Q) of order p; 0 if x ∈ G \ CG(Q) of order greater than p. The map β is a class function because we have the following sequence of subgroups each normal in G: 0 < Z(G) < Q < CG(Q) < G. For the sequel of the proof, we set φk : Z/pn → C; x 7→ e2piikx/n while φkφj : Z/pn × Z/pn → C; (x, y) 7→ e2pii(kx+jy)/n. To lighten the notation, we drop the dependence in n on the φk’s because it will be clear from the context. To understand correctly the characters that follow, it will be important to specify the generators of the elementary abelian subgroups treated. Consider first an elementary abelian subgroup H of G of rank 2 and which intersects trivially the center Z(G). If H ∩Q 6= 0, then H ∩Q ∼= Z/p and H ∼= (H ∩Q)× Z/p so that: β|H = |G| p−1∑ k=0 p−1∑ j=1 φkφj 24 2.3. Some p-groups If H ∩Q = 0, then Z/p ∼= H ∩CG(Q) so that H ∼= (H ∩CG(Q))×Z/p and: β|H = (p− 1)|G|/p p−1∑ k=0 p−1∑ j=1 φjφk + |G| p−1∑ k=1 φ0φk. Consider now a subgroup H of G with H ∩Z(G) = 0. We will proceed case by case using the classification above. 1. If H ∩Q 6= 0 then H ⊂ CG(Q) and |K| = p with K = Q ∩H. Let φ be the character of K which is p − 1 on the identity and −1 for each other element of K. Then: β|H = p|G||H : K|Ind H Kφ. 2. If H∩Q = 0 and H ⊂ CG(Q) then H is cyclic and β|H = |G|/|H|(p2− p)φ where φ is the character of H that is |H| on the identity and 0 elsewhere. 3. If H ∩Q = 0 and H is cyclic with H ∩ CG(Q) = 0, then |H| = p and β|H = |G|/pφ where φ is (p3 − p2) on the identity and −p elsewhere. 4. If H ∩ Q = 0 and H is cyclic with H ∩ CG(Q) 6= 0, then β|H = (p2 − p)|G|/|H|∑|H|k=1 φk. 5. Assume that H ∩ Q = 0 and that H is abelian of type (p, pn−1). Write H =< x, y ∈ H|xp = ypn−1 = 1, [x, y] = 1 >. Notice that < y >= H ∩CG(Q). For each 1 ≤ i ≤ p set Hi =< xyipn−2 >. Clearly |Hi| = p, Hi ∩ CG(Q) = 0 and Hi ∩Hj = 0 if i 6= j. Let φi be the character of Hi which is p − 1 on the identity and −1 elsewhere. Set: φ = p∑ i=1 IndHHiφi. Since φ(1) = |H|(p−1), φ(z) = −|H|/p for z ∈ H\ < y > and φ(z) = 0 for z ∈< y >; we conclude that β|H = p|G|/|H|φ. 25 2.3. Some p-groups 6. If H ∩Q = 0 and H ∼= M(pn), we can write H =< x, y|xpn−1 = yp = 1, y−1xy = x1+pn+2 >. Let N =< xpn−2 , y >∼= (Z/p)2 which is normal in H. Let φ be the character of N given by: β|H = (p− 1) p−1∑ i=0 p−1∑ j=1 φjφi + p p−1∑ i=1 φ0φi. Then we have φ(0) = p2(p−1), φ((xkpn−2 , 0)) = 0, while φ((xkpn−2 , yl)) = −p. Finally β|H = IndHN |G|/p|H : N |φ. We can now turn our attention to the topological problem: Proposition 2.3.4. For every odd order rank 3 p-group G, there is a finite dimensional G-CW-complex X ' Sm × Sn with cyclic isotropy subgroups. Proof. If Z(G) is not cyclic, then it is enough to consider the linear spheres of representations of G induced from free representations of some Z/p×Z/p < Z(G). Assume that Z(G) is cyclic and let Sm be the linear G-sphere obtained by inducing from a free linear action of Z(G). The isotropy subgroups for this action are the one described in proposition 2.3.2. The conditions of proposition 2.2.6 are fulfilled by proposition 2.3.3. The conclusion follows. As a direct consequence of theorem 3.2 in [2] we obtain: Theorem 2.3.5. For every odd order rank 3 p-group G, there is a free finite G-CW-complex X ∼= Sm × Sn × Sk. A converse to theorem 2.3.5 is given by Hanke in [23] in the sense that: if (Z/p)r acts freely on X = Sn1 × ...×Snk and if p > 3dim(X), then r ≤ k. 26 2.3. Some p-groups Remark 2.3.6. For p = 2 the situation is more complicated because of the classification of subgroups. A 2-group of rank 1 can be either cyclic or generalized quaternion. A 2-group with a maximal abelian subgroup can be cyclic, generalized quaternion, dihedral, M(2n) (see proposition 2.3.2) or S4m =< x, y|x2m = y2 = 1, y−1xy = xm−1 > (see [37] section 4 chapter 4). For p = 2, the class function of proposition 2.3.3 does not restrict to characters over the subgroups, in general. Next, we present another family of p-groups for which we can recover a free action on the desired product of spheres. This is the family of central extensions of abelian p-groups. We first prove a stronger result, a general- ization of theorem 3.2 in [2], and then we specialize it to central extensions of abelian p-groups. Theorem 2.3.7. Let G be a finite p-group and let X be a finite dimensional G-CW-complex with Gσ abelian for all cells σ ⊂ X. Then there is a free finite dimensional G-CW-complex Y ' X × Sn1 × ...× SrkX(G). Moreover, if X is finite, then Y is finite as well. Proof. We prove the theorem by induction over rkX(G). If rkX(G) = 1, the theorem has been proven by A. Adem and J. Smith (3.2 in [2]). The inductive step follows. By virtue of proposition 2.2.6, we only need to associate to each isotropy subgroup Gσ a representation ρσ : Gσ → U(m) such that ρσ|Gτ ∼= ρτ whenever Gτ < Gσ and such that ρσ is fixed point free for all Gσ with rk(Gσ) = rkX(G). Consider the class function β : G→ C given by: x 7→  |G|(prkX(G) − 1), if x = 0; −|G|, if o(x) = p; 0, otherwise. To simplify the notation write A = Gσ for an isotropy subgroup (which is abelian by hypothesis). We need to prove that β|A is a character which is fixed point free whenever A ∼= (Z/p)rkX(G). Set Ap = {0}∪{x ∈ A|o(x) = p}. 27 2.4. Infinite groups Since A is abelian we have Ap / A. Fix an injection f : Ap → (Z/p)rkX(G). Write ρ0 : (Z/p)rkX(G) → U(prkX(G) − 1) for the reduced regular repre- sentation and let ρ = ρ0 ◦ f be the representation Ap → (Z/p)rkX(G) → U(prkX(G) − 1). Consider finally the representation of A given by η = |G||Ap|/|A|IndAApρ. Clearly η(0) = |G|(prkX(G)−1), η(x) = −|G| if x ∈ Ap\0 while η(x) = 0 if x /∈ Ap. As a result β|A = η. Let now A ∼= (Z/p)rkX(G). Clearly β|A is a multiple of the reduced regular representation, thus fixed point free. Theorem 2.3.8. Let G be a p-group. Assume that G is a central extension of abelians, then there is a free finite G-CW-complex X ' Sn1× ...×Snrk(G). The result in particular holds for extraspecial p-groups. Proof. Let X = Sn1× ...×Snrk(Z(G)) be the product of the G-spheres arising from suitable representations of the center. Clearly rkX(G) = rk(G) − rk(Z(G)) and Gσ is abelian. The conclusion follows. Remark 2.3.9. For extraspecial p-groups, similar results have been obtained independently by Űnlű and Yalçin [44]. 2.4 Infinite groups As pointed out in [16], there is a class of infinite groups which is worth considering, when studying the rank conjecture. This is the class of groups Γ with finite virtual cohomological dimension. Recall that, by definition, a group Γ has finite virtual cohomological dimension, if it has a finite in- dex subgroup Γ′ < Γ with finite cohomological dimension (that is to say: Hn(Γ′) = 0 for all coefficients and for all n big enough). Writing vcd for virtual cohomological dimension and cd for cohomological dimension, one can show that the number vcd(Γ) = cd(Γ′) is well defined. See for example [10] for background on groups with finite virtual cohomological dimension. The crucial property that makes them interesting to us is the following: for any such group Γ there exists a finite dimensional Γ-CW-complex EΓ 28 2.4. Infinite groups with |Γx| <∞ for all x ∈ EΓ. It is already known that a group with finite virtual cohomological di- mension, which is countable and with rank at most one finite subgroups, acts freely on a finite dimensional CW-complex X ' Sm [16]. The next step would be to prove the analogue result for groups Γ with rank at most two finite subgroups. The easiest examples to consider are amalgamated products Γ = G1 ∗G0 G2, where Gi is a finite group for i = 0, 1, 2 and G0 < Gj for j = 1, 2. In this case, for every finite subgroup H < Γ, there is γ ∈ Γ such that γHγ−1 < Gi for i = 1 or i = 2 (see [34]). In particular rk(Γ) = max {rk(G1), rk(G2)}. The first attempt would be to find an effective Γ-sphere, i.e. a Γ-sphere with rank at most one isotropy subgroups. We exhibit now an amalgamation of two p-groups which doesn’t have an effective Γ-sphere. Theorem 2.4.1. For p an odd prime, there is an infinite group Γ with fi- nite virtual cohomological dimension, satisfying the two following properties: every finite subgroup G < Γ is a p-group with rk(G) ≤ 2 and for every finite dimensional Γ-CW-complex X ' Sn there is at least one isotropy subgroup Γσ with rk(Γσ) = 2. Proof. Let E and E′ be two copies of the extraspecial p-group of order p3 and exponent p. (Such a group can be identified with the upper triangular 3× 3 matrices over Fp with 1 on the diagonal). Consider the amalgamated product Γ = E′ ∗Z/pE given by Z/p = Z(E) and an injective map f : Z/p→ E with f(Z/p) ∩ Z(E′) = 1. Clearly rk(Γ) = 2. Let Γ act on a finite dimensional CW-complex X ' Sn. Consider the restriction of this action to E and E′. It is well known that the dimension function of a p-group action on a sphere is realized by a representation over the real numbers [18]. Therefore, an even multiple of the dimension functions for E and E′ must be realized by characters χE and χE′ . Clearly the dimension functions of χE and χE′ must agree over Z(E) and f(Z/p). Looking at the character table of E, we observe that every 29 2.4. Infinite groups irreducible character α, giving rise to an effective sphere, vanishes outside Z(E) while α(z) = mζp for all z ∈ Z(E)\{0} (here ζp is a p-root of the unity). Thus, χE and χE′ cannot be both characters giving rise to effective spheres. We deduce that the original action must have some finite isotropy subgroups of rank 2. This provides an example of an infinite group, with rank 2 finite p-subgroups, not acting with effective Euler class on any sphere. Remark 2.4.2. This kind of behaviour cannot happen with finite groups: every rank 2 finite p-group, has a linear sphere with periodic isotropy sub- groups. Let’s try to approach the rank conjecture algebraically. For a finite group G, we have that rk(G) = r if and only if there are r finite dimensional Z [G]- complexes K1, ...,Kr such that K = K1⊗ ...⊗Kr is a complex of projective Z [G]-modules with H∗(K) ∼= H∗(Sn1 × ...× Snr) (see [7]). In corollary 2.4.6 we prove a similar result: for every group Γ with vcd(Γ) <∞, there are rk(Γ) finite dimensional Z [Γ]-complexesC1, ...,Crk(Γ) such that D = C∗(EΓ) ⊗C1 ⊗ ... ⊗Crk(Γ) is a complex of projective Z [Γ]- modules with H∗(D) ∼= H∗(Sn1 × ...× Snrk(Γ)). As a result, the group Γ introduced in theorem 2.4.1 satisfies the alge- braic analogue of the rank conjecture but does not have an effective Γ-sphere. The geometric problem of knowing whether or not Γ acts freely on a product of two spheres is still open. We begin by recalling some preliminaries concerning the cohomology of finite groups. We follow here [8] and [9]. Let G be a finite group. Consider ζ ∈ Hn(G,R) ∼= ExtnRG(R,R) ∼= HomRG(Ω̂nR,R), where Ω̂nR is the nth kernel in a RG-projective resolution P of R. We choose a cocycle ζ̂ : Ω̂nR→ R representing ζ. By making P large enough we can assume that ζ̂ is 30 2.4. Infinite groups surjective. We denote Lζ its kernel and form the pushout diagram: Lζ = //  Lζ  0 // Ω̂nR // ζ̂  Pn−1 //  Pn−2 //  ... // P0 //  R // =  0 0 // R // Pn−1/Lζ // Pn−2 // ... // P0 // R // 0 We denote by Cζ the chain complex: 0→ Pn−1/Lζ → Pn−2 → ...→ P0 → R→ 0 formed by truncating the bottom row of this diagram. Thus we have that H0(Cζ) = Hn−1(Cζ) = R while Hi(Cζ) = 0 if i 6= 0, n − 1. A useful result is given in the proof of theorem 3.1 in [3]: Proposition 2.4.3. Let G be a finite group. For all positive integer r, there exist classes ξ1, ..., ξr ∈ H∗(G,Z) such that, for all H < G with rk(H) ≤ r, the complex Z [G/H]⊗ Lξ1 ⊗ ...⊗ Lξr is Z [G]-projective. Proof. See the proof of theorem 3.1 in [3] Corollary 2.4.4. Let G be a finite group. For all positive integer r, there exist finite dimensional Z [G]-complexes C1, ...,Cr such that H∗(C1 ⊗ ... ⊗ Cr) = H∗(Sn1 × ...× Snr); with C1 ⊗ ...⊗Cr a complex of Z [H]-projective modules for all H < G with rk(H) ≤ r. Proof. Let ξ1, ..., ξr ∈ H∗(G,Z) be the classes given in proposition 2.4.3. Consider the chain complex C = Cξ1⊗ ...⊗Cξr . Clearly H∗(C) = H∗(Sn1× ... × Snr). For the second part of the claim, observe that all the modules in Cξi are Z [G]-projective except the module Pni−1/Lξi . Recall that the tensor product of any module with a projective module is projective, so that it remains to examine the module Pn1−1/Lξ1 ⊗ ... ⊗ Pnr−1/Lξr . Let 31 2.4. Infinite groups H < G be such that rk(H) ≤ r. Since Z [G/H] ⊗ Lξ1 ⊗ ... ⊗ Lξr is Z [G]- projective by proposition 2.4.3, we conclude that Z [G/H] ⊗ Pn1−1/Lξ1 ⊗ ... ⊗ Pnr−1/Lξr is Z [G]-projective as in 5.14.2 of [9]. It then easily follows that Pn1−1/Lξ1 ⊗ ...⊗ Pnr−1/Lξr is Z [H]-projective. This is the end of the reminder about the cohomology of finite groups. We now apply it to the case of an infinite group Γ with vcd(Γ) < ∞ and rank r. Write Γ′ for a torsion-free normal subgroup of Γ with G = Γ/Γ′ finite. We apply corollary 2.4.4 to Γ/Γ′ with r = rk(Γ). We recover a Z [Γ]- complex C1 ⊗ ... ⊗Cr such that H∗(C1 ⊗ ... ⊗Cr) = H∗(Sn1 × ... × Snr); with C1⊗ ...⊗Cr a complex of Z [H]-projective modules for all finite H < Γ. On the other hand, we have that the ZΓ-complex C∗(EΓ) is contractible. Therefore the complex D = C∗(EΓ) ⊗C1 ⊗ ... ⊗Cr is such that H∗(D) = H∗(Sn1 × ...× Snr). Lemma 2.4.5. With the notation above, the complex D is Z [Γ]-projective. Proof. The complex C∗(EΓ) decomposes as a graded direct sum of permu- tation modules: C∗(EΓ) = ⊕σZ [Γ/Γσ] = ⊕σZ [Γ] ⊗Z[Γσ ] Z. Here σ spans the cells of EΓ/Γ and the grading is given by the dimensions of the cells σ. Consequently D = ⊕σ(Z [Γ] ⊗Z[Γσ ] C1 ⊗ ... ⊗Cr), so that we only need to prove that Z [Γ]⊗Z[Γσ ]C1 ⊗ ...⊗Cr is Z [Γ]-projective. Let Qσ be a graded Z [Γσ]-module such that (C1⊗...⊗Cr)⊕Qσ is Z [Γσ]-free. We then have that (Z [Γ]⊗Z[Γσ ]C1⊗...⊗Cr)⊕(Z [Γ]⊗Z[Γσ ]Qσ) = Z [Γ]⊗Z[Γσ ]((C1⊗...⊗Cr)⊕Qσ) is Z [Γ]-free. Consequently, the algebraic version of the rank conjecture holds for groups of finite virtual cohomological dimension: Corollary 2.4.6. For a group Γ with vcd(Γ) <∞ and rk(Γ) = r, there exist a finite dimensional contractible complex C∗(EΓ) and r finite dimensional Z [Γ]-complexes C1,...,Cr such that D = C∗(EΓ)⊗C1 ⊗ ...⊗Cr is a Z [Γ]- projective complex with H∗(D) ∼= H∗(Sn1 × ...× Snr). 32 2.4. Infinite groups Remark 2.4.7. We thank professor D. Benson for pointing out to us, that a similar result has already been proved in [27]. 33 Chapter 3 Finite Homotopy G-Spheres up to Borel Equivalence In this chapter we discuss the study of finite homotopy G-spheres up to Borel equivalence. That is to say, given a homotopy G-sphere X, we want to know if it is Borel equivalent to a finite one. As explained by the classification theorems of Grodal and Smith (here theorems 1.0.1 and 1.0.2), Borel equiva- lences are captured by dimension functions and quasi-isomorphisms of some chain complexes over the orbit category. In the first section we introduce the reader to the topic of homological algebra over orbit categories. In the second section, we use it to present a new approach to the construction of finite homotopy G-spheres. In particular, we give new examples for groups of the form Cp o Cqr . 3.1 Homological algebra over the orbit category The goal of this section is to introduce the reader to the theory of homo- logical algebra over the orbit category. The main reference for this section is [29]. We need to begin by discussing what we mean by modules over the orbit category. In general, one may define the notion of modules over any category, and most of the results presented below hold for modules over any finite, ordered, E.I. category. Since we will only be concerned with the modules over the orbit category (which is, in particular, a finite ordered E.I. category) we will not present the following results in the most general setting. 34 3.1. Homological algebra over the orbit category Let G be a finite group and F a family of subgroups closed under the actions of conjugation and taking subgroups. The orbit category of G with respect to the family F , denoted OrFG, has transitive G-sets as objects and G-maps as morphisms. More concretely, we may write the objects of the orbit category as the set: Ob(OrFG) = {G/H|H ∈ F} the set morphisms between two G orbits is given by: Mor(G/H,G/K) = {g ∈ G|Hg < K} /K. In what follows, we will write Γ = OrFG for the full orbit category and Γp for the orbit category OrFG, where F is the family of subgroups of prime power order for some prime p dividing the order of G. Notice that Γp is a full subcategory of Γ. Let R be a commutative ring with unit. An RΓ-module M is a contravariant functor: M : Γop → R−Mod from the orbit category to the category of R-modules. A morphism of RΓ- modules is then a natural transformation of functors. The category of RΓ modules is denoted by RΓ−Mod. Since Γ is a small category and R−Mod is abelian, the category of RΓ-modules is abelian as well. We are able, therefore, to do homological algebra in the category RΓ−Mod. For RΓ- modules, the terms exact, injective, surjective, etc. are determined object wise. For instance, the sequence of RΓ-modules: L→M → N is exact if and only if the sequence of R-modules: L(G/H)→M(G/H)→ N(G/H) 35 3.1. Homological algebra over the orbit category is exact for all G/H ∈ ObΓ. As usual, a RΓ-module P is projective if and only if the functor: HomRΓ(P,−) : RΓ−Mod→ R−Mod is exact. For an orbit G/H ∈ ObΓ, define the free module generated at G/H by: FG/H(G/K) = RMor(G/K,G/H) for all G/K ∈ ObΓ. Here RMor(G/K,G/H) is the free R-module on the set of RΓ morphisms from G/K to G/H. The RΓ-module FG/H is defined in such a way that maps of RΓ-modules FG/H →M are determined by the image of IdG/H ∈ FG/H(G/H) in M(G/H). In particular: HomRΓ(FG/H ,M) ∼=M(G/H) which says that FG/H is a projective RΓ-module for each G/H ∈ ObΓ. We are now able to give a definition of a free RΓ-module: Definition 3.1.1. An RΓ-module M is called free if it is isomorphic to:⊕ G/H∈λ FG/H for some collection of orbits λ ⊂ ObΓ. Just as in the case of modules over a ring, one can prove that a RΓ- module is projective if and only if it is a summand of a free module. Simi- larly, we say that a RΓ-module M is finitely generated if there is a finitely generated free RΓ-module covering it: ⊕ki=1FG/Hi  M . A chain complex is called perfect if it is a finite dimensional, finitely generated chain complex of projective modules. Next, we prove a result of homological algebra that will be needed in the following section. Lemma 3.1.2. Every perfect free QΓ-chain complex C is quasi-isomorphic to its homology H(C). 36 3.1. Homological algebra over the orbit category Proof. A QΓ-complex C is a family of Q-complexes with maps between them. Clearly, each of these complexes is quasi-isomorphic to its homology as a Q-complex. In order to motivate what follows, we recall why this is the case: Let D be a perfect Q-chain complex. Using a scalar product in each degree, we can write D under the form: 0→ ...→ Dn = Ker(dn)⊕Ker⊥(dn)→ ...→ 0 The projection map D → H(D) is then well defined and it is a quasi- isomorphism. In order for the same method to work in the case of a QΓ- complex, we want to be able to split the Q-modules Cn(G/H) in a functorial way. To this end, it is enough to prove that the maps Cn(G/K)→ Cn(G/L) respect the scalar products. This is what we are going to show next. Since C is QΓ-free, all the Q-modules involved must necessarily be of the form QMor(G/H ′, G/H), for some H ′,H < G. Such a vector space comes with a canonical basis with respect to which we define a canonical scalar product. To study Cn(G/K) → Cn(G/K), observe that all the self-maps of the module QMor(G/K,G/H) preserve such a canonical scalar product, be- cause they are given by permutation matrices induced by maps of the form g : Mor(G/K,G/H) → Mor(G/K,G/H); u 7→ ug for g ∈ WK = NK/K, the Weil group. To study Cn(G/K) → Cn(G/L), notice that all maps QMor(G/K,G/H)→ QMor(G/L,G/H) respect the canonical scalar products because they are given by set injections g : Mor(G/K,G/H) → Mor(G/L,G/H); u 7→ ug for elements g ∈ Mor(G/L,G/K). (Each mor- phism g ∈ Mor(G/L,G/K) being surjective makes the corresponding map g :Mor(G/K,G/H)→Mor(G/L,G/H) injective). To summarize: we showed that for a perfect free QΓ-complex C, there are QΓ-modules Kern and Ker⊥n such that Cn = Kern ⊕Ker⊥n . Moreover, Kern(G/H) = Ker(Cn(G/H) → Cn−1(G/H)). As a consequence, we can define a map of QΓ-complexes C → H(C) which is a quasi-isomorphism. 37 3.1. Homological algebra over the orbit category We end by recalling the notation of obstruction theory for RΓ-complexes (see [29]). Let K0(RΓ) be the Groethendieck group of finitely generated pro- jective RΓ-modules. Let F be subgroup of K0(RΓ) generated by free RΓ- modules. The reduced K-theory group, is the quotient K̃0(RΓ) = K0(RΓ)/F . For a perfect RΓ-complex P , the sum o(P ) = ∑∞ i=1(−1)iPi defines a class õ(P ) ∈ K̃0(RΓ). If P is a perfect complex and if f : P → C is a quasi-isomorphism, then we set õ(C) := õ(P ). It is the obstruction for C to be free in the sense of proposition 11.2 in [29]: Proposition 3.1.3. . With the above notation, õ(C) = 0 if and only if C is weakly equivalent to a perfect free RΓ-complex. If C is perfect, then õ(C) = 0 if and only if C is homotopy equivalent to a perfect free RΓ-complex. We would like to be able to express the obstruction theory for RΓ- modules in terms of the obstruction theories of R [WH]-modules, forH some subgroups of G. The trouble here is that the functor ResH : [RΓ−Mod]→ [R [WH]−Mod]; M 7→M(G/H) does not take projective to projective! We need a better functor: For a RΓ-modules M , following [29], let M(G/H)s be the R-submodule of M(G/H) generated by all images of R- homomorphisms M(f) : M(G/K) → M(G/H) induced by all non isomor- phisms f : G/H → G/K. Observe that M(G/H)s is directly a R [WH]- submodule because for f ∈WH, the composition g ◦ f is an isomorphism if and only if g is. Finally, we define the splitting functor SH : [RΓ−Mod]→ [R [WH]−Mod] by M 7→ M(G/H)/M(G/H)s. As in [29], one can then prove that: Proposition 3.1.4. The splitting functors SH respect ”direct sum”, ”finitely generated”, ”free” and ”projective”. To compute the obstruction class, we now have theorem 10.34 of [29]: Theorem 3.1.5. The reduced K-theory splits as: K̃0(RΓ) ∼= ⊕(H)<GK̃0(R [WH]). 38 3.2. Finite homotopy G-spheres Here the sum runs over the conjugacy classes of subgroups H < G. Further- more, if P∗ is a perfect chain complex, then the above isomorphism sends õ(P∗) to ⊕(H)<Gõ(SH(P∗)). 3.2 Finite homotopy G-spheres Let X be a homotopy G-sphere. Recall that for all p||G| and for all p- subgroupsK, we have that the homotopy fixed pointsXhK =MapK(EK,X) have the mod p homology of a sphere. In particular, for such a space, we have a family of dimension functions { DimpX(−) } p||G|. For us, each Dim p X(−) is defined over Γp by Dim p X(K) = Dim(H ∗(XhK ,Z(p))). These dimension functions are preserved under Borel equivalence, since Y hK ' XhK , if X and Y are two Borel equivalent G-spheres. Remark 3.2.1. Elsewhere [14], [21], dimension functions have been defined by DimpX(K) = Dim(H ∗(XhK ,Z(p))) + 1. Such a definition behaves better with respect to join products and reduced homology. For our realization purposes, we find it more convenient to use a different convention. As usual DimpX(K) = −1 means that XhK = ∅. Given a homotopy G-sphere X, we want to know if we can find a per- fect free ZΓ-complex C, whose homology realizes the dimension functions{ DimpX(−) } p||G|. If such a chain complex exists, we want to know if we can realize it as the chain complex of a space Y , which will be our finite model for the original sphere X. The next example shows that it is convenient to first look for perfect free Z(p)Γ-complexes Cp that realize the p-dimension function DimpX(−). The example also prepares the terrain for new and more general results, see theorem 3.2.6. Example 3.2.2. This example is a mix of example 4.4.2 and lemma 4.3.5 in [14]. Consider the symmetric group G = Σ3 = C3oC2 =< σ, τ |σ3 = τ2 = 1, τστ = σ2 >. We order the category Γ as follows: (G/G,G/C3, G/C2, G/e). We want to find chain complexes D3 ∈ Z(3)Γ and D2 ∈ Z(2)Γ realizing the dimension function (−1,−1, 3, 3). 39 3.2. Finite homotopy G-spheres We begin at the prime 3. We will need the fact that the number 2 is invertible in Z(3), later in the construction. Notice that H∗(Σ3,Z(3)) is two periodic, so that, by [38], there is an exact chain complex: 0 // Z(3) //M3 f3 //M2 f2 //M1 f1 //M0 // Z(3) // 0 with Mi a perfect free Z(3)Σ3-module for i = 0, ..., 3. Since the free module generated at G/e is FG/e = (0, 0, 0,Z(3)Σ3), we recover the following free Z(3)Γ-complex D′3: [3] M3 f3  [2] M2 f2  [1] M1 f1  [0] M0 Σ3/Σ3 Σ3/C3 Σ3/C2 Σ3/e Its dimension function is (−1,−1,−1, 3). Not quite what we want. Observe first that the map of G-sets pi : G→ G/C2 ∼= C3 induces an exact sequence of Z(3)Σ3-modules: 0 // Ker(pi) // Z(3)Σ3 pi // Z(3)C3 // 0 . The map µ : Z(3)C3 → Z(3)Σ3, given by [ (σi, τ j) ] 7→ (σi, τ0) defines a Z(3)C3-splitting of the exact sequence 5. As usual, since 2 is invertible in Z(3), we can define a Z(3)Σ3-splitting of the exact sequence, by setting µG : Z(3)C3 → Z(3)Σ3, [ (σi, τ j) ] 7→ ((σi, τ0) + (σi, τ1))/2. In particular, the free 5C3 is the subgroup of Σ3 generated by σ, while C3 is the Σ3-set given by Σ3/C2 40 3.2. Finite homotopy G-spheres module generated at G/e splits as FG/e = (0, 0, 0,Z(3)C3 ⊕Ker(pi)). Recall also that the free Z(3)Γ-module generated at G/C2 is (0, 0,Z(3),Z(3)C3). We can now modify D′3 into a perfect free Z(3)Γ-complex D3, which re- alizes the dimension function (−1,−1, 3, 3). To describe the new complex, we use the symbol ρi to designate the projection of a direct sum into its i-th component. We leave out the columns Σ3/Σ3 and Σ3/C3 because they are trivial: [4] Z(3)C3 ⊕Ker(pi) ρ1⊕0⊕ρ2⊕0  [3] Z(3) 0  Z(3)C3 ⊕ Z(3)C3 ⊕Ker(pi)⊕M3 0⊕ρ2⊕0⊕(f3◦ρ4)  [2] Z(3) 1  Z(3)C3 ⊕ Z(3)C3 ⊕Ker(pi)⊕M2 ρ1⊕0⊕ρ3⊕(f2◦ρ4)  [1] Z(3) 0  Z(3)C3 ⊕ Z(3)C3 ⊕Ker(pi)⊕M1 ρ2⊕(f1◦ρ4)  [0] Z(3) Z(3)C3 ⊕M0 Σ3/C2 Σ3/e Next, we construct a free Z(2)Γ-complex D2, with the required homology. Again, we will need 3 to be invertible in Z(2) for the construction to hold. 41 3.2. Finite homotopy G-spheres Consider first the perfect free Z(2)Γ-complex D′2: [2] Z(2)[Σ3] f2  [1] Z(2) 0  Z(2)C3 ⊕ Z(2)[Σ3] f1◦ρ2  [0] Z(2) Z(2)C3 Σ3/Σ3 Σ3/C3 Σ3/C2 Σ3/e The maps f1 and f2 are given by: f1(e) = σ − σ2 f2(e) = (σ, e− στ − σ2τ). Over the orbits G/Σ3, G/C3 and G/C2, everything is clear. We need to study the Z(2)Σ3-complex sitting over the orbit G/e. It is easy the check that Im(f1 ◦ ρ2) = Ker( : Z(2)[Σ3/C2] → Z(2)), so that H0 = (0, 0,Z(2),Z(2)). Some elementary linear algebra, shows that we have H1 = (0, 0,Z(2),Z(2)) because the coefficients are in Z(2). More in detail, after a suitable change of basis, the matrix of f2 : Z(2)Σ3 → Ker(f1 ◦ ρ2) is given by: 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0  Since 9 is invertible in Z(2), we have that Ker(0 + f1)/Im(f2) ∼= Z(2) and Ker(f2) = 0. Finally, by concatenating D′2 with itself, we obtain the desired 42 3.2. Finite homotopy G-spheres perfect free Z(2)Γ-complex D2 realizing (−1,−1, 3, 3). Similar observations and examples were already made by Clarkson in [14], where he used Postnikov towers to glue the p-local complexes together in the case of groups of the form G = CpoCq. His strategy relies on the fact that |G| = pq. In order to extend Clarkson’s result (see theorem 3.2.6), we suggest a gluing method based on an arithmetic square, which does not need the condition |G| = pq. We will still need Postnikov towers in order to show that the output of the arithmetic square satisfies some properties. Following [22], we recall the main result concerning Postnikov towers of RΓ-complexes: Given a projective RΓ chain complex C, there is a sequence of projective chain complexes C(i) together with maps fi : C → C(i) inducing homology isomorphisms for dimension ≤ i. Moreover, there is a tower of maps: C(i)  C(i− 1)    αi // Σi+1P (Hi) C EE                  ;;xxxxxxxxxx // ##F FF FF FF FF F C(1)  α2 // Σ3P (H2) C(0) α1 // Σ2P (H1) such that C(i) = Σ−1C(αi), where C(αi) denotes the algebraic mapping cone of αi and P (Hi) denotes a projective resolution of the homology module Hi. Recall also that the algebraic mapping cone of a chain map f : C → D is defined as the chain complex C(f) = D⊕ΣC with boundary map ∂(x, y) = (∂(x) + f(y), ∂y). Note that Σn is the shift operator for chain complexes given by (ΣnC)i = Ci−n. It then follows that: Proposition 3.2.3. Let C be a projective chain complex with finite homo- logical dimension and such that Hi(C) is finitely generated for all i. If there is an integer n such that ExtiRΓ(C,M) = 0 for all modules M and i > n, 43 3.2. Finite homotopy G-spheres then C is homotopy equivalent to an n-dimensional, finitely generated pro- jective chain complex. Proof. Let d be the homological dimension of C and consider the d-th stage of the Postnikov tower for C. By inspection of the above description of Post- nikov towers, C(d) is a finitely generated projective chain complex. The map fd : C → C(d) is a homology isomorphism, so that C and C(d) are homo- topy equivalent. As a consequence, ExtiRΓ(C(d),M) = 0 for all modules M and i > n. By lemma 6.1 in [22], C(d) is then homotopy equivalent to an n-dimensional, finitely generated projective chain complex. We can now state our gluing theorem: Theorem 3.2.4. Consider a finite family of primes P = {p1, ..., pn}. As- sume that for every prime pi ∈ P we are given a perfect free Z(pi)Γ-complex Dpi such that H(Dpi⊗Q) is isomorphic to H(Dpj⊗Q) for all i and j. If there is a perfect free Z(1/P )Γ-complex D1/P such that H(D1/P ⊗Q) is isomorphic to H(Dpj ⊗ Q), then there is a perfect free ZΓ-complex D with D ⊗ Z(pi) quasi-isomorphic to Dpi for all pi ∈ P and D ⊗ Z(1/P ) quasi-isomorphic to D1/P . Proof. Consider the limit: D = lim uukkkk kkkk kkkk kkkk k yysss sss sss sss  %%KK KKK KKK KKK K ))SSS SSSS SSSS SSSS SS Dp1 ))SSS SSS SSS SSS SSS S ... %%JJ JJ JJ JJ JJ J D1/P  ... yyttt tt tt tt tt Dpn uukkkk kkk kkk kkk kkk H(Dp1 ⊗Q) Observe that the maps into H(Dp1⊗Q) are given by lemma 3.1.2. As usual, we will use the flatness of Q to move the coefficients out of the homology. Assume first that P = {p}. The limit is then a pullback giving rise to a 44 3.2. Finite homotopy G-spheres short exact sequence: 0→ D → Dp ⊕D1/p → H(Dp)⊗Q→ 0 which in turn yields a long exact sequence: ...→ Hn(D)→ Hn(Dp ⊕D1/p)→ Hn(Dp)⊗Q→ ... Applying the exact functor −⊗ Z(p), gives an exact sequence: ...→ Hn(D)⊗Z(p) → (Hn(Dp)⊕Hn(D1/p))⊗Z(p) → Hn(Dp)⊗Q⊗Z(p) → ... Since Q⊗Z(p) = Q = Z(1/p)⊗Z(p), the map Hn(D1/p)⊗Z(p) → Hn(Dp)⊗Q is surjective for all n, hence: Hn(D)⊗ Z(p) ∼= Ker [ (Hn(Dp)⊕Hn(D1/p))⊗ Z(p) → Hn(Dp)⊗Q ] Such a kernel is nothing but the pullback: Hn(Dp) //  Hn(Dp)  Hn(D1/p)⊗Q ∼= // Hn(Dp)⊗Q We readily conclude that Hn(D)⊗Z(p) ∼= Hn(Dp). Similarly Hn(D)⊗Z(q) ∼= Hn(D1/P )⊗ Z(q) for all q /∈ P , so that Hn(D)⊗ Z(1/P ) ∼= Hn(D1/P ), by the very essence of arithmetic squares. For the general case it is enough to considerD as a sequence of pullbacks, and apply the above argument repeatedly. Next we have to prove that D is quasi-isomorphic to a perfect ZΓ- complex using 3.2.3. For that purpose, we need a well known algebraic fact [26]: a Z-module M is finitely generated (resp. finite or trivial), if and only if all the M ⊗ Z(pi) and M ⊗ Z(1/P ) are finitely generated (resp. finite or trivial) Z(pi) and Z(1/P ) modules. In our case, it immediately follows 45 3.2. Finite homotopy G-spheres that D has finite homological dimension and Hi(D) is finitely generated for all i. Up to taking a projective resolution of D, we can assume that we have a projective chain complex. In particular, we can assume that all the quasi-isomorphisms above are homotopy equivalences. To apply proposition 3.2.3, we also need to study the groups ExtkZΓ(D,M) for all ZΓ-modules M . As Z(pi) is flat over Z, we have that Ext k ZΓ(D,M)⊗ Z(pi) = Ext k Z(pi)Γ (D ⊗ Z(pi),M ⊗ Z(pi)) = ExtkZ(pi)Γ(Dpi ,M ⊗ Z(pi)). By assumption, Dpi is perfect so that Ext k Z(pi)Γ (Dpi ,M ⊗ Z(pi)) = 0 for all k > dim(Dpi) and for all pi ∈ P . Similarly, we have that ExtkZΓ(D,M) ⊗ Z(1/P ) = ExtkZ(1/P )Γ(D1/P ,M ⊗ Z(1/P )) = 0 for all k > dim(D1/P ) As a result, ExtkZΓ(D,M) = 0 for all k big enough. By 3.2.3, D is equivalent to a perfect chain complex. Now that we know that D is quasi-isomorphic to a perfect ZΓ-complex, we can talk about its obstruction õ(D) ∈ K̃0(ZΓ). Recall that õ(D)⊗Z(pi) = õ(D ⊗ Z(pi)) = õ(Cpi) for all pi ∈ P and of course õ(D) ⊗ Z(1/P ) = õ(D ⊗ Z(1/P )) = õ(D1/P ). This implies that D is quasi-isomorphic to a projective free ZΓ-complex, provided that all the complexes Dpi and D1/P are free. Now that we can build perfect free ZΓ-complexes with prescribed homol- ogy, we need a way of realizing them. This is done in theorem 8.10 of [22], for which we need some definitions: We call a function n : Ob(OrFG)→ N monotone if n(G/K) ≤ n(G/H) whenever (H) < (K). An ROrFG-complex C is called an n-Moore complex, if it is connected and if H̃i(C(G/H)) = 0 for i 6= n(G/H). Theorem 3.2.5. [22] Let Γ = OrFG and consider a perfect free ZΓ-complex C. Suppose that C is an n-Moore complex with n(G/H) ≥ 3 for all G/H ∈ Ob(Γ). Suppose further that Ci(G/H) = 0 for all i ≥ n(G/H) + 1 and for all G/H ∈ Ob(Γ). Then there is a finite G-CW-complex X such that C(X) is homotopy equivalent to C as a ZΓ-complex. Moreover Gx ∈ F for all x ∈ X. 46 3.2. Finite homotopy G-spheres Now that we have explained the general strategy to build finite homotopy G-spheres, we are ready for some examples. Systematizing the methods of example 3.2.2 and using proposition 3.2.4, we obtain the following new result: Theorem 3.2.6. Consider the group G = CpoCqr with faithful Cqr action on Cp. For all s ≤ r and for all j ≥ 3, there is a finite homotopy G-sphere X with: DimqX(G/Cqt) = { j + 2qr if t ≤ s, j otherwise. while DimpX(G/Cp) = j. We prove first two technical lemmas. Lemma 3.2.7. Let G be the group Cp o Cqr . For each s ≤ r there is a perfect free Z(p)Γ-complex Tp such that: Hi(Tp(G/K)) = { Z(p) if e 6= (K) < (Cqs), 0 otherwise. for i = 0, 1, while Hi(Tp) = 0 if i > 1. Proof. The free Z(p)Γ-functor generated at G/Cqs has value: FG/Cqs (G/K) =  Z(p)Cqr−s if e 6= (K) < (Cqs), Z(p)Cp o Cqr−s if e = K, 0 otherwise. Consider the map FG/Cqs → FG/Cqs induced by the group ring norm map N : Z(p)Cqr−s → Z(p)Cqr−s , t 7→ t− t2; where t is a generator of Cqr−s . This gives rise to a Z(p)Γ-complex T ′p with: Hi(T ′p(G/K)) =  Z(p) if e 6= (K) < (Cqs), Z(p)Cp if e = K, 0 otherwise. 47 3.2. Finite homotopy G-spheres for i = 0, 1 and Hi(T ′) = 0 for i > 1. In the same fashion as example 3.2.2, one can prove that there exists a Z(p)G-module M such that Z(p)G ∼= Z(p)Cp⊕M , as Z(p)G-modules. Following the construction of example 3.2.2 again, one can easily define a perfect free Z(p)Γ-complex Tp: Z(p)Cp ⊕M // Z(p)Cp ⊕M ⊕ FG/Cqs // FG/Cqs with the required homology. Lemma 3.2.8. The Z(q)G-module Ker( : Z(q)Cp → Z(q)) is projective. Proof. The short exact sequence of coefficients: 0 // Ker() // Z(q)Cp  // Z(q) // 0 induces a long exact sequence in homology: ... // Hn(G,Ker()) // Hn(G,Z(q)Cp) ∼= // Hn(G,Z(q)) // ... SinceHn(G,Z(q)) = Hn(Cqr ,Z(q)) = Hn(G,Z(q) [G/Cqr ]), we must have that Hn(G,Ker()) = 0, so that Ker() is a projective Z(q)G-module. Here is the proof of theorem 3.2.6 Proof. In analogy to example 3.2.2, we begin by finding perfect free chain complexes Dp ∈ Z(p)Γ and Dq ∈ Z(q)Γ with the required homologies. Con- sider first the prime p. By [38], there is a chain complex D′p: 0 // Z(p) //M2qr−1 f2qr−1 // ... f1 //M0 // Z(p) // 0 withMi a perfect free Z(p)G-module for all i = 0, ..., 2qr−1. We can look at this chain complex as a perfect free Z(p)Γ-complex, with zeroes away from the orbit G/e. Let T ∗2q r p be the concatenation of 2qr copies of the complex Tp given in lemma 3.2.7. The required chain complex Dp is then found by 48 3.2. Finite homotopy G-spheres concatenating T ∗2q r p ⊕D′p with the trivial algebraic G-sphere of dimension j given by the chain complex: Ci(Sj) = { FG/G = Z(p) if i = 0, j; 0 otherwise. We treat next the case of the prime q. Consider again the map FG/Cqs → FG/Cqs induced by the norm map N : Z(q)Cqr−s → Z(q)Cqr−s . As in lemma 3.2.7, it defines a chain complex L, with homology: Hi(L(G/K)) =  Z(q) if e 6= (K) < (Cqs), Z(q)Cp if e = K, 0 otherwise. for i = 0, 1 and Hi = 0 for i > 1. We add now free modules to modify this chain complex and reduce the homology at G/e from Z(q)Cp to Z(q): Using lemma 3.2.8, the Z(q)G-module Q = Ker( : Z(q)Cp → Z(q)) is summand of some free Z(q)G-module F = Q ⊕ B. We can then easily produce a chain complex D′q of the form F → F ⊕ FG/Cqs → FG/Cqs with homology: Hi(D′q(G/K)) = { Z(p) if (K) < (Cqs), 0 otherwise. for i = 0, 1 andHi(D′q) = 0 for i > 1 The required perfect free Z(q)Γ-complex Dq is found again by concatenating (D′) ∗2qr q with the trivial algebraic G- sphere of dimension j. We want next to apply proposition 3.2.4 to the complexes Dp and Dq. Observe that any of the constructions for either Dp or Dq, equally defines a perfect free Z(1/pq)Γ complex D1/pq realizing the given dimension function. We conclude that 3.2.4 provides a perfect free ZΓ-complex D realizing the same dimension function. Navigating through the various constructions, we observe that the di- mensional requirements of 3.2.5 are satisfied. Finally, we find a finite G- 49 3.2. Finite homotopy G-spheres CW-complex X with C(X) quasi-isomorphic to D, by virtue of theorem 3.2.5. Since X is finite, the fixed points XK are mod-p equivalent to the homotopy fixed points XhK for all p-subgroups K and all p||G|. As a re- sult, X is our finite G-sphere, Borel equivalent to all G-spheres Y with{ DimpX(−) } p||G| = { DimpY (−) } p||G|. 50 Bibliography [1] A. Adem, J. Milgram. Cohomology of finite groups, Springer, (1994). [2] A. Adem, J. Smith. Periodic complexes and group actions, Ann. of Math. (2) 154 (2000), 407-435. [3] A. Adem. Torsion in equivariant cohomology, Comm. Math. Helv. 64 (1989), 401-411. [4] A. Adem. Lectures on the cohomology of finite groups, Contemporary Mathematics 436 (2007), 317-340. [5] A. Adem, W. Browder. The free rank of symmetry of (Sn)k, Invent. Math. 92, (1988), 431-440. [6] M. Barratt, V. Gugenheim, J. Moore. On semisimplicial fiber- bundles, Amer. J. Math. 81 (1959), 639-657. [7] D. Benson, J.Carlson. Complexity and multiple complexes, Math. Zeit. 195 (1987), 221-238. [8] D. Benson. Representations and cohomology I, Cambridge University Press, (1991). [9] D. Benson. Representations and cohomology II, Cambridge University Press, (1991). [10] K.S. Brown Cohomology of groups, Springer-Verlag, (1982). [11] G. Bredon. Equivariant cohomology theory, Lecture Notes in Mathe- matics 34, (1967) 51 Bibliography [12] G. Carlsson. On the rank of abelian groups acting freely on (Sn)k, Invent. Math 69, (1982), 393-400. [13] H. Cartan, S. Eilenberg. Homological algebra, Princeton University Press, (1956). [14] J. Clarkson. Group actions on finite homotopy spheres, Phd Thesis, UBC, 2010. [15] P.E. Conner. On the action of a finite group on Sn×Sn, Ann. Math. 66, (1957). [16] F. Connolly, S. Prassidis. Groups which acts freely on Rm × Sn, Topology 28 (1989), 133-148. [17] C. Curtis, I. Reiner. Methods of representation theory II, Wiley In- terscience, (1987) [18] R. Dotzel, G. Hamrick. p-Groups actions on homology spheres., Inventiones Mathematicae 62 (1981), 437-442. [19] D. Ferrario. Self homotopy equivalences of equivariant spheres, Groups of homotopy self-equivalences and related topics (1999), 105- 131. [20] D. Ferrario. Self-equivalences of dihedral spheres, Collectanea Math- ematica 53 (3), (2002), 251-264. [21] J. Grodal, J. Smith. Classification of homotopy G-actions on spheres, In preparation, 2010. [22] I. Hambleton, S. Pamuk, E. Yalçin. Equivariant CW-complexes and the orbit category, arXiv:0807.3357v2, 2008. [23] B. Hanke. The stable free rank of symmetry of products of spheres, Invent. Math. 178 (2) (2009), 265-298. [24] A. Hatcher Algebraic topology, Cambridge university press, (2002). 52 Bibliography [25] A. Heller. A note on spaces with operators, Illinois Journal of Math- ematics 3 (1959), 98-100. [26] P. Hilton, G. Mislin, J. Roitberg Localization of nilpotent groups and spaces, Mathematics studies, 15, (1975). [27] J. H. Jo. Multiple complexes and gaps in Farrell cohomology, Journal of pure and applied algebra 194, (2004), 147-158. [28] M. Jackson. Rank three p-groups and free actions on the homotopy product of three spheres, In preparation. [29] W. Lueck Transformation groups and algebraic K-theory, Lecture notes in mathematics, 1408, (1989). [30] I. Madsen, C.B. Thomas, C.T.C. Wall. The topological spherical space form problem II, Topology 15 (1978), 375-382. [31] J. Milnor. Groups which acts on Sn without fixed points, Amer. J. Math. 79 (1957), 623-630. [32] R. Oliver. Free compact group actions on products of spheres, Springer-Verlag LNM 763 (1978), 539-548. [33] S. Prassidis. Groups with infinite virtual cohomological dimension which act freely on Rm × Sn., Journal of pure and applied algebra 78 (1992), 85-100. [34] J.P. Serre. Cohomologie des groupes discrets, Ann. Math. Stud. 70 (1971), 77-169. [35] P.A. Smith. Permutable periodic transformations, Proc. Nat. Acad. Sci. 30 (1944), 105-108. [36] N. Steenrod. The topology of fibre bundles, Princeton University Press, (1951). [37] M. Suzuki. Group theory II, New York, (1986). 53 Bibliography [38] R. Swan. Periodic resolutions for finite groups, Annals of Mathematics 78 (1960), 267-291. [39] R. Swan. Induced representations and projective modules, The annals of mathematics, Vol.71 n.3, (1960), 552-578. [40] T. Tom Dieck. Transformation groups and representation theory, Lec- ture Notes in Mathematics 766, (1979). [41] T. Tom Dieck. Transformation groups, de Gruyter Studies in Math- ematics, (1987). [42] T. Tom Dieck, T. Petrie. Homotopy representations of finite groups, Publications Mathematiques de l’IHES 56, (1982), 129-169. [43] O. Űnlű. Constructions of free group actions on products of spheres, Phd Thesis, University of Wisconsin, Madison, (2004). [44] O. Űnlű, E Yalçin. Fusion systems and constructing free actions on products of spheres, arxiv. 54


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