{"http:\/\/dx.doi.org\/10.14288\/1.0080542":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Mathematics, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Klaus, Michele","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2011-07-14T16:48:05Z","type":"literal","lang":"en"},{"value":"2011","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Doctor of Philosophy - PhD","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"In the first part of the thesis we discuss the rank conjecture of Benson and Carlson. 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 homotopy equivalent to the product of rk(G) spheres; where rk(G) is the rank of G.\n\nWe 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 L satisfying the two following properties: every finite subgroup G 3dim(X), then r \u2264 k. Another way of extending Swan\u2019s 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 \u0393 acts freely and properly on Rn \u00d7 Sm if and only if \u0393 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 \u0393 with finite virtual cohomo- logical dimension 3. The reason is that for every such group \u0393, there is a contractible finite dimensional \u0393-CW-complex E\u0393, 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 \u0393 with finite virtual cohomological dimension, satisfying the two following proper- ties: every finite subgroup G < \u0393 is a p-group with rk(G) \u2264 2 and for every finite dimensional \u0393-CW-complex X ' Sn there is at least one isotropy subgroup \u0393\u03c3 with rk(\u0393\u03c3) = 2. 3We say that \u0393 has finite virtual cohomological dimension if there is \u0393\u2032 < \u0393 with |\u0393\/\u0393\u2032| finite and Hn(\u0393\u2032) = 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 \u2192 Y and g : Y \u2192 X and G-equivariant homotopies F : X \u00d7 I \u2192 X from g \u25e6f to IdX and G : Y \u00d7I \u2192 Y from f \u25e6g to IdY . Under this definition, the G-spaces EG and \u2217 are not equivariantly homotopy equivalent because there is no equivariant map \u2217 \u2192 EG. On the other hand, there is a G-equivariant map EG\u2192 \u2217 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 \u00d7G X and EG \u00d7G 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 \u2192 Z1 \u2190 ...\u2192 Zk \u2190 Y each of which is a homotopy equivalence. Clearly, in this setting, EG is equivalent to \u2217. 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 \u0393 be the orbit category OrG. This is the category with Ob(\u0393) = {G\/H|H < G} and Mor(G\/K,G\/H) = {f : G\/K \u2192 G\/H|f is a G-equivariant map}. For a prime p, the category \u0393p, is the full subcategory of \u0393 defined by: G\/H \u2208 Ob(\u0393p) 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(\u2212) } 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(\u2212) : \u0393p \u2192 N; G\/K 7\u2192 Dim(H\u2217(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(\u2212) = DimpY (\u2212) for all p||G|. Moreover, every family of functions {Dp(\u2212) : \u0393p \u2192 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(\u2212) satifies the Borel Smith condition for all p||G|; 3. Dp(\u2212) 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(Fp\u0393p) for the category of contravariant functors F : \u0393opp \u2192 C(Fp). Theorem 1.0.2. [21] The family of functors {\u03a6p}p||G| defined by \u03a6p : {homotopy G-spheres} \u2192 C(Fp\u0393p), with \u03a6p(X)(G\/K) = C\u2217(MapG(EG \u00d7 G\/K,X),Fp), satisfies the following properties: 1. For all X, the chain complex \u03a6p(X) is quasi-isomorphic to a perfect Fp\u0393p-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 \u03a6p(Y ) and \u03a6p(X) are quasi-isomorphic for all p||G|, if and only if X and Y are Borel equivalent. 4A function Dp(\u2212) : \u0393p \u2192 N respects fusion, if Dp(K) = Dp(K\u2032) whenever there is g \u2208 G with gKg\u22121 = K\u2032. We say that Dp satisfies the Borel-Smith condition if Dp(\u2212)|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)\u2212DimpX(G\/Cp). His method involves Dold\u2019s theory of algebraic Postnikov towers and relies on the assumption that |G| = pq. Inspired by Clarkson\u2019s 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 \u2264 r and for all j \u2265 3, there is a finite homotopy G-sphere X with: DimqX(G\/Cqt) = { j + 2qr if t \u2264 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 \u2265 0, there exists a positive integer q such that the group pik(AutG(S(V \u2295q))) 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\u00d7S1\u00d7...\u00d7Snk . 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 \u2208 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 \u2192 E, x 7\u2192 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 \u2192 X is G-equivariant. The r-th skeleton of a pair (X,A) is the space Skr(X,A) = Skr(X) \u222a 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\u22121), we recover a chain complex of ZG modules C\u2217(X,A): ... \/\/ Hn+1(Xn+1, Xn) \/\/ Hn(Xn, Xn\u22121) \/\/ ... If M is another ZG-module, the cochain complex: C\u2217G(X,A;M) = HomZG(C\u2217(X,A),M) yields cohomology groups H\u2217G(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\u22121 as a pushout: \u2210 i\u2208J G\u00d7 Sn\u22121 \/\/ \u000f\u000f Xn\u22121 \u000f\u000f\u2210 i\u2208J G\u00d7Dn \/\/ Xn The corresponding characteristic map: \u03c6 : \u2210 i\u2208J G\u00d7 (Dn, Sn\u22121)\u2192 (Xn, Xn\u22121) provides a canonical basis for the free ZG-module Hn(Xn, Xn\u22121). It is given by the images of the canonical generators of Hn(Dn, Sn\u22121) under the j-th component of \u03c6: {e} \u00d7 (Dn, Sn\u22121) \/\/ G\u00d7 (Dn, Sn\u22121) \u03c6j \/\/ (Xn, Xn\u22121) . 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) \u223c= 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) \u2192 [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\u2217G(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 \u2265 1 there exists an exact obstruction sequence: [Xn+1, Y ]G \u2192 Im([Xn, Y ]G \u2192 [Xn\u22121, Y ]G) c n+1\u2212\u2212\u2212\u2192 Hn+1G (X,A;pinY ) which is natural in (X,A) and Y . The exactness of this sequence means that each homotopy class Xn\u22121 \u2192 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\u22121 \u2192 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 \u03c6 : \u2210 i\u2208J G \u00d7 (Dn+1, S1) \u2192 (Xn+1, Xn) be the characteristic map and ej \u2208 C\u2217(X,A) be the basis element corresponding to the j-th component \u03c6j . Fix an element h \u2208 [Xn, Y ]G. The composition h\u25e6\u03c6j defines an element cn+1(h)(ej) \u2208 [Sn, Y ] = pinY . Extending by linearity, we recover an element cn+1(h) \u2208 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\u22121, Y ], then cn+1(h0) and cn+1(h1) differ by a coboundary. For that purpose, choose a G-homotopy k : I\u00d7Xn\u22121 with ki = hi|Xn\u22121 . Suppose that \u03d5 : (Dn, Sn\u22121)\u2192 (X,Xn\u22121) is the characteristic map of an n-cell defining a basis element e \u2208 Cn(X,A). By composition, we recover a map: {0} \u00d7Dn \u222a I \u00d7 Sn\u22121 \u222a {1} \u00d7Dn \u03d5\u00d7Id \/\/ {0} \u00d7Xn \u222a I \u00d7Xn\u22121 \u222a {1} \u00d7Xn (h0,k,h1) \u000f\u000f Y 10 2.1. Equivariant obstruction theory Composing with the standard homeomorphism Sn \u223c= \u2202I \u00d7Dn \u222a I \u00d7 Sn\u22121, yields a homotopy class x \u2208 [Sn, Y ] = pinY . Setting d(h0, k, h1) : Cn(X,A)\u2192 pinY ; e 7\u2192 x we recover an element d(h0, k, h1) \u2208 CnG(X,A;pinY ). It turns out that \u03b4d(h0, k, h1) = cn+1(h0)\u2212 cn+1(h1) as required. As a by-product, the end of the proof also provides an obstruction cochain d(h0, k, h1) \u2208 CnG(X,A;pinY ), defined for two maps h0, h1 : X \u2192 Y and a G-homotopy k : I\u00d7Xn\u22121 \u2192 Xn\u22121, with ki = hi|Xn\u22121 . 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 d\u0304(h0, k, h1) \u2208 HnG(X,A;pinY ). Moreover: 1. d\u0304(h0, k, h1) = \u2212d\u0304(h1, k\u2212, h0), where k\u2212 is the inverse homotopy, 2. d\u0304(h0, k, h1) + d\u0304(h1, k\u2032, h2) = d\u0304(h0, k + k\u2032, h2), 3. d\u0304(h0, k, h1) = 0 if and only if the G homotopy k extends to a G- homotopy K : I \u00d7Xn \u2192 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 \u2282 ... \u2282 Xm given by Xi = {x \u2208 X| (Gx) = (Hj) for some j \u2264 i}. Such a filtration allows us to re- cover pairs (XHii , X H1 i\u22121) with free WHi = NHi\/Hi-action on X Hi i \\ XHii\u22121. Moreover: Proposition 2.1.4. [40] The WHi = NHi\/Hi-action on XHii \\XHii\u22121 is free. Furthermore, given a G-map k : Xi\u22121 \u2192 Yi\u22121, the extensions K : Xi \u2192 Yi of f , are in bijective correspondence with the WHi-extension e : XHii \u2192 Y Hii of k : XHii\u22121 \u2192 Y Hii\u22121. 11 2.1. Equivariant obstruction theory Proof. Set XHi = {x \u2208 X| (Gx) = (Hi)}. Given K, we have e = KHi and since GXHi = Xi \\ Xi\u22121, the G map K is uniquely determined by KHi . Which shows injectivity. Conversely, suppose that we are given a WHi-map e : XHii \u2192 Y Hii extending kH . We define a map E : Xi \u2192 Yi by: x 7\u2192 { e(x), if x \u2208 Xi\u22121; ge(y), if x = gy with y \u2208 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 \u2192 Y . Assume that there is i0 such that f1|Xi0 : Xi0 \u2192 Yi0 and f2|Xi0 : Xi0 \u2192 Yi0 are homotopic. In order to know if f1 and f2 are homotopic, we want to know if the successive restrictions f1|Xi : Xi \u2192 Yi and f2|Xi : Xi \u2192 Yi are homotopic for all i > i0. Since homotopies are G-maps F : I \u00d7 X \u2192 Y with trivial G-action on I, we can consider (I \u00d7 X)i = I \u00d7 Xi. A G-extension K : I \u00d7 Xi \u2192 Yi of the given G-homotopy k : I \u00d7Xi\u22121 \u2192 Yi\u22121, exists if and only if there is a WHi-extension of kHi : I \u00d7XHii\u22121 \u2192 Y Hii\u22121, by proposition 2.1.4. By theorem 2.1.3, such a WHi-extension exists, if and only if the dif- ference cocycles d\u0304((f1)i, kHi , (f2)i) \u2208 HnG(XHii , XHii\u22121;pinYi) are all zero, for n = 1, ..., Dim(XHii , X Hi i\u22121). By virtue of remark 2.1.1, this homology groups satisfy: HnG(X Hi i , X Hi i\u22121;pinYi) \u223c= Hn(XHii \/WHi, XHii\u22121\/WHi;pinYi), where pinYi is interpreted as a local coefficients system over XHii \\XHii\u22121. 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\u22121 = \u222aH>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 \u2192 X which is an homotopy equivalence and for which there exists an homotopy inverse g : X \u2192 X and homotopies F,H : X \u00d7 I \u2192 X, between f \u25e6 g and IdX , and between IdX and g \u25e6 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\u22121 to Dn. Thus we want to study the groups [ Sn\u22121, BAutG(Sm) ] = pin\u22121(BAutG(Sm)) = pin\u22122(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 \u2208 AutG(X) we write AutG(X)f for the path component of f . For k > 0, the map of unbased homotopy classes \u03d5 : [ Sk, AutG(X)f ] \u2192 [Sk \u00d7X,X] G is injective and factors through: [ Sk, AutG(X)f ] \u000f\u000f \u03d5 \/\/ [ Sk \u00d7X,X] G pik(AutG(X)f ) 66mmmmmmmmmmmmm 13 2.2. A general construction In particular all G-equivariant homotopies H : I\u00d7Sk\u00d7X \u2192 X between maps representing the same element in Im(\u03d5) can be taken to satisfy H(t, \u2217, x) = H(t\u2032, \u2217, x) for all t, t\u2032 \u2208 I and x \u2208 X. Proof. The map \u03d5 : [ Sk, AutG(X)f ] \u2192 [Sk \u00d7X,X] G is clearly well de- fined. To see that it is injective, consider a G-equivariant homotopy H : I\u00d7Sk\u00d7X \u2192 X from \u03d5(g1) to \u03d5(g2). ClearlyH|{0}\u00d7{x0}\u00d7X = \u03d5(g1)(x0,\u2212) = g1(x0) \u2208 AutG(X)f . Which implies that H|{t}\u00d7{x}\u00d7X \u2208 AutG(X)f for all (t, x) \u2208 I \u00d7 Sk because H|{t}\u00d7{x}\u00d7X ' H|{0}\u00d7{x0}\u00d7X via a path in I \u00d7 Sk from (0, x0) to (t, x). As a result, H defines an homotopy from g1 to g2. To prove that \u03d5 factors through: [ Sk, AutG(X)f ] \u000f\u000f \u03d5 \/\/ [ Sk \u00d7X,X] G pik(AutG(X)f ) 66mmmmmmmmmmmmm we want to show that the map pik(AutG(X)f ) \u2192 [ 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 \u2192 AutG(X)f with g 7\u2192 f \u25e6g. Conse- quently pi1(AutG(X)f ) acts trivially on pik(AutG(X)f ) for all f \u2208 AutG(X). We conclude that pik(AutG(X)f )\u2192 [ 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 \u2208 X| Gx \u2208 (H1)}, where (H1) denotes the conjugacy class of H1. We then have that AutG(X1) \u223c= AutWH1(XH1) (here WH1 = NH1\/H1 is the Weil group). Proof. Let\u2019s begin by studying X1. Clearly X1 \u2282 \u222aH\u2208(H1)XH . Since H \u2208 (H1) is supposed to be maximal, we must have that if x \u2208 XH , then Gx = H 14 2.2. A general construction so that X1 = \u222aH\u2208(H1)XH . Similarly, if x \u2208 XH \u2229XH \u2032 , for H, H \u2032 in (H1), then H = Gx = H \u2032. As a result X1 = \u222aH\u2208(H1)XH . Observe next that a G-equivariant map f : X1 \u2192 X1 restricts to aWH1- equivariant map f1 : XH1 \u2192 XH1 because WH1 = NH1\/H1 and H1 acts trivially on X1. The same holds for a G-equivariant homotopy F : I\u00d7X1 \u2192 X1, so that we have a well defined map res : AutG(X1)\u2192 AutWH1(XH1). One can then show that the map res : AutG(X1) \u2192 AutWH1(XH1) has an inverse given by res\u22121(f)(x) = gf(g\u22121x), where g \u2208 G is such that g\u22121x \u2208 XH1 . Lemma 2.2.3. Let G be a finite group and Sn a linear G-sphere. If 0 < k < n then Hn(Sk \u00d7 Sn\/G, {\u2217} \u00d7 Sn\/G,Z) is finite. Proof. Consider the long exact sequence of the pair (Sk\u00d7Sn\/G, {\u2217}\u00d7Sn\/G) with integer coefficients: Hn\u22121(Sk \u00d7 Sn\/G) \/\/ \/\/ Hn\u22121({\u2217} \u00d7 Sn\/G) \u000f\u000f Hn(Sk \u00d7 Sn\/G) i\u2217 \u000f\u000f Hn(Sk \u00d7 Sn\/G, {\u2217} \u00d7 Sn\/G)oo Hn({\u2217} \u00d7 Sn\/G) Clearly Hn(Sk \u00d7 Sn\/G, {\u2217} \u00d7 Sn\/G) \u2282 Ker(i\u2217). But Hn(Sk \u00d7 Sn\/G) \u223c= Hn({\u2217} \u00d7 Sn\/G) \u2295 Hn\u2212k({\u2217} \u00d7 Sn\/G). Thus for i\u2217 : Hn(Sk \u00d7 Sn\/G) \u2192 Hn({\u2217} \u00d7 Sn\/G) we have that Ker(i\u2217) \u223c= Hn\u2212k({\u2217} \u00d7 Sn\/G). Finally, the groups Hn\u2212k(Sn\/G) are finite for 0 < k < n because Hn\u2212k(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 \u2264 G write nr(H) for the integer such that S(V \u2295r)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\u00d7Snq(Hi)\/WHi,\u222aH>HiSk\u00d7Snq(H)\/WHi\u222a{?}\u00d7Snq(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) \u2212 nr,i > 0. Observe that nr(H) = rn1(H) + (r \u2212 1) so that nr,i = rn1,i + (r \u2212 1) and mr,i = nr(Hi)\u2212nr,i = rn1(Hi)+(r\u22121)\u2212(rn1,i+(r\u22121)) = rm1,i. Therefore there is a qi big enough such that mr,i > k+2. In other words nr(Hi)\u2212k\u22122 > nr,i. We have found an integer qi > 0 such that all the cells \u03c4 of the CW-complex Snqi (Hi) of dimension dim(\u03c4) \u2265 nqi(Hi)\u2212 k\u2212 2, are also cells of the relative CW-complex (Snqi (Hi),\u222aH>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 \u03c4 of the CW- complex Sk\u00d7Snqi (Hi)\/WHi of dimension dim(\u03c4) \u2265 nqi(Hi)\u22122, are also cells of the relative CW-complex (Sk\u00d7Snqi (Hi)\/WHi,\u222aH>HiSk\u00d7Snqi (H)\/WHi). Henceforth: Hnqi (Hi)(Sk \u00d7Snqi (Hi)\/WHi,\u222aH>HiSk \u00d7Snqi (H)\/WHi \u222a {?}\u00d7 Snqi (Hi),Z) = Hn(Hi)(Sk\u00d7Snq1 (Hi)\/WHi, {?}\u00d7Snqi (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 0 such that pik(AutG(S(V \u2295q))) 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 \u2282 ... \u2282 S(V )m = S(V ) given by S(V )i = {x \u2208 S(V )| (Gx) = (Hj); j \u2264 i}. We have homomorphisms Ri : pik(AutG(S(V ))) \u2192 pik(AutG(S(V )i)) and Si : pik(AutG(S(V )i)) \u2192 pik(AutG(S(V )i\u22121)). 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 \u001c\u001c8 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 pik(AutG(S(V )m)) \u000f\u000f... Si+1 \u000f\u000f pik(AutG(S(V )i)) Si \u000f\u000f... S2 \u000f\u000f 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\u22121i (Ri\u22121(f)) \u2229 Im(Ri) is finite for all i and for all f \u2208 pik(AutG(S(V ))). This outline can only be carried out up to replacing S(V ) with some power S(V \u2295q). We begin by showing that there is q1 > 0 such that pik(AutG(S(V \u2295q1)1)) is finite. In particular we will have that Im(R1) \u2282 pik(AutG(S(V \u2295q1)1)) is finite. For H < G write nr(H) for the integer such that S(V \u2295r)H = Snr(H). Observe that nr(H) = rn(H) + (r \u2212 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)\u2212 1. If k \u2265 n1(H1)\u22121, then there is a q1 > 0 for which k < q1n1(H1)+(q1\u22121)\u22121 = nq1(H1) \u2212 1. As a result pik(AutG(S(V \u2295q1)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\u22121) is finite. The inductive step is to prove that there is q \u2265 qi\u22121 such that S\u22121i (Ri\u22121(f)) \u2229 Im(Ri) is finite for all i and for all f \u2208 pik(AutG(S(V \u2295q))). 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\u2032 > 0 be such that the groups: Hnq\u2032 (H \u2032)(Sk\u00d7Snq\u2032 (H\u2032)\/WH \u2032,\u222aH>H\u2032Sk\u00d7Snq\u2032 (H)\/WH \u2032\u222a{?}\u00d7Snq\u2032 (H\u2032)\/WH \u2032) are finite for all H \u2032 < G with n1(H \u2032) > 0. Let q = max {q1, q\u2032}. To simplify the notation we write W = V \u2295q, X = Sk \u00d7 S(W ) and X\u0304Hi = \u222aH>HiXH \u222a {?} \u00d7 S(W )Hi . With this notation we have that the group: Hnq(Hi)(XHi\/WHi, X\u0304Hi\/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, X\u0304Hi\/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 X\u0304Hi (see end of section 2.1). We have a map over the part of X\u0304Hi given by \u222aH>HiXH , because of the inductive hypothesis. We have a map over the part of X\u0304Hi given by {\u2217} \u00d7 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)) \u2192 18 2.2. A general construction [ Sk \u00d7 S(W )i, S(W )i ] G yielding a diagram with injective columns: pik(AutG(S(W )i)) Si \/\/ \u03d5i \u000f\u000f pik(AutG(S(W )i\u22121)) \u03d5i\u22121 \u000f\u000f [Xi, S(W )i]G si \/\/ [Xi\u22121, S(W )i\u22121]G To prove that S\u22121i (Ri\u22121(f)) \u2229 Im(Ri) is finite, it is enough to prove that s\u22121i (\u03d5i\u22121(Ri\u22121(f)))\u2229\u03d5i(Im(Ri)) is finite. By abuse of notation we will keep on writing Si and Ri\u22121(f), but we will think of them as living in the bottom row of the diagram. Now, a homotopy h between Ri\u22121(f) and Ri\u22121(g) is con- stant over {\u2217}\u00d7S(W )i\u22121 where it coincides with both Ri\u22121(f) and Ri\u22121(g). Consequently, h can be extended to a homotopy from Ri(f)|Xi\u22121\u222a{\u2217}\u00d7S(W )i to Ri(g)|Xi\u22121\u222a{\u2217}\u00d7S(W )i . Henceforth, writing Xi\u22121 \u222aS(W )i for Xi\u22121 \u222a{\u2217}\u00d7S(W )i, we can apply equivariant obstruction theory inductively over r to each of the diagrams: [Skr+1(Xi, Xi\u22121 \u222a S(W )i), S(W )i]G Si,r+1 \/\/ Skr \u000f\u000f [Xi\u22121, S(W )i\u22121]G [Skr(Xi, Xi\u22121 \u222a S(W )i), S(W )i]G Si,r 33fffffffffffffffffffffff If r = 0, then Sk0(Xi, Xi\u22121 \u222a S(W )i) = Xi\u22121 \u2210 {x0, ..., xl}. Consequently, S\u22121i,0 (Ri\u22121(f)) \u2229 Sk0(Im(Ri)) depends on the connected components of the space S(W )i\u22121. But S(W )i\u22121 has finitely many connected components because it is a finite CW-complex, therefore S\u22121i,0 (Ri\u22121(f)) \u2229 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\u22121i,r (fi\u22121)\u2229Skr(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 \u2208 S\u22121i,r+1(fi\u22121) \u2229 Skr+1(Im(Ri)) there is a unique gji,r and a homo- topy h from gi,r = gi,r+1|Skr(Xi,Xi\u22121\u222aS(W )i) to gji,r Notice that, by definition, we have gj , g \u2208 pik(AutG(S(W ))) with gi,r+1 = gi|Skr+1(Xi,Xi\u22121\u222aS(W )i) and gji,r = g j i |Skr(Xi,Xi\u22121\u222aS(W )i). 19 2.2. A general construction We write d\u0304(gi,r, h, g j i,r) \u2208 Hr+1(XHi\/WHi, X\u0304Hi\/WHi, pir+1((Sn)Hi)) for the homology class of the difference cocyle as in the preceding section. Now, if d\u0304(g\u2032i,r+1, h \u2032, Skr+1(g j i )) = d\u0304(gi,r+1, h, Skr+1(g j i )), then we have: d\u0304(g\u2032i,r+1, h \u2032+h\u22121, gi,r+1) = d\u0304(g\u2032i,r+1, h \u2032, Skr+1(g j i ))+ d\u0304(Skr+1(g j i ), h \u22121, gi,r+1) = d\u0304(g\u2032i,r+1, h \u2032, Skr+1(g j i ))\u2212 d\u0304(gi,r+1, h, Skr+1(gji )) = 0 so that gi,r+1 ' g\u2032i,r+1. We can therefore define an injection: S\u22121i,r+1(fi\u22121) \u2229 Skr+1(Im(Ri)) \u000f\u000f\u2210t j=1 { (gji,r) } \u00d7Hr+1(XHi\/WHi, X\u0304Hi\/WHi, pir+1(Snq(Hi))) by setting gi 7\u2192 { gji } \u00d7 d\u0304(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\u22121i,r+1(fi\u22121) \u2229 Skr+1(Im(Ri)) is finite for all r. Since X is finite dimensional, this shows that S\u22121i (fi\u22121)\u2229 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 \u201dattach\u201d 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 \u00d7 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 \u2208 N| there exists G\u03c3 with rk(G\u03c3) = 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 G\u03c3 we can associate a representation \u03c1\u03c3 : G\u03c3 \u2192 U(n) such that \u03c1\u03c3|G\u03c4 \u223c= \u03c1\u03c4 when- ever G\u03c4 < G\u03c3. If \u03c1\u03c3 is fixed point free for all G\u03c3 with rk(G\u03c3) = rkX(G), then there exists a finite dimensional G-CW-complex E \u223c= X \u00d7 Sm with rkE(G) = rkX(G)\u2212 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\u22121\u03c3 for the linear sphere associated to \u03c1\u03c3. 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 \u03c3 and define a mapG\u00d7G\u03c3S2n\u22121\u03c3 \u2192 X0 by (g, x) 7\u2192 g\u00b7\u03c3. This defines a G-equivariant spherical fibration S2n\u22121 \u2192 E0 \u2192 Sk0(X) whose total space is a finite dimensional G-CW-complex. Clearly if \u03c1\u03c3 is fixed point free for all G\u03c3 with rk(G\u03c3) = rkX(G), then rkE0(G) = rkX(G)\u2212 1. The inductive step is next. Suppose given a G-equivariant spherical fibration over the (k \u2212 1)-skeleton \u2217qk\u22121S2n\u22121 \u2192 Ek\u22121 \u2192 Skk\u22121(X) whose total space is a finite dimensional G-CW-complex. Assume also that if \u03c1\u03c3 is fixed point free for all G\u03c3 with rk(G\u03c3) = rkX(G), then rkEk\u22121(G) = rkX(G) \u2212 1. Now, for every G-orbit of a k-cell, choose a representative \u03c3. The G\u03c3-equivariant fibration \u2217qk\u22121S2n\u22121 \u2192 Ek\u22121|\u2202\u03c3 \u2192 \u2202\u03c3 is classified by an element a\u03c3 \u2208 pik\u22122(AutG\u03c3(\u2217qk\u22121S2n\u22121)). We want to have a\u03c3 = 0: Observe that, in general, for two complex G-spheres S(V ) and S(W ), we have that S(V \u2295 W ) \u223c= S(V ) \u2217 S(W ) as G-spheres. Therefore, by proposition 2.2.5, we can take enough Whitney sums of the fibration \u2217qk\u22121S2n\u22121 \u2192 Ek\u22121 \u2192 Skk\u22121(X) to guarantee that a\u03c3 = 0 (see lemma 2.3 and proposition 2.4 in [16]). We can then extend the G\u03c3-equivariant fibration \u2217qkS2n\u22121 \u2192 Ek\u22121|\u2202\u03c3 \u2192 \u2202\u03c3 equivariantly across the cell \u03c3. We define a G-equivariant spherical fibration over the orbit of \u03c3 by 21 2.2. A general construction G\u00d7G\u03c3 \u2217qkS2n\u22121\u03c3 \u2192 G\u03c3 with (g, x) 7\u2192 g \u00b7 \u03c3. Repeating the procedure for all the representatives of the G-orbits of k-cells, we recover a G-equivariant spherical fibration \u2217qkS2n\u22121 \u2192 Ek \u2192 Skk(X) with total space a finite dimensional G-CW-complex. Clearly if \u03c1\u03c3 is fixed point free for all G\u03c3 with rk(G\u03c3) = rkX(G), then rkEk(G) = rkX(G)\u22121. We conclude noticing that, by proposition 2.8 in [2], up to taking further fiber joins, we can assume that the total fibration \u2217qS2n\u22121 \u2192 E \u2192 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 \u00d7 ... \u00d7 Snt is a finitely dimensional G-CW-complex with rkX(G) = rk(G) \u2212 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 \u00d7 ... \u00d7 Snt \u00d7 Snt+1 with rkY1(G) = rk(G) \u2212 (t + 1). Clearly, we then wish to apply proposition 2.2.6 again and again until we recover a suitable free space Yrk(G)\u2212t ' Sn1 \u00d7 ...\u00d7 Snt \u00d7 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 \u00d7 ... \u00d7 Snt where each Sni is the linear sphere of a representation \u03c1i of Z(G). Consider the induced representations \u03b7i = IndGZ(G)\u03c1i. We recover a G-space X = S m1 \u00d7 ... \u00d7 Smt , where Smi is the linear G-sphere corresponding to \u03b7i and rkX(G) = rk(G)\u2212 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 \u2229 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 \u2229 Z(G) = 0 and |H| = pn. Then either H is cyclic, H < CG(Q), H is abelian of type (p, pn\u22121) or H \u223c= M(pn) =< x, y|xpn\u22121 = yp = 1, y\u22121xy = x1+pn\u22122 >. Proof. If rk(H) = 1 then H is cyclic since p > 2. Suppose that rk(H) = 2 and H \u2229 Q 6= 0. By assumption Z(G) \u2229 Q = Z\/p, H \u2229 Z(G) = 0 and Q \u2229 H = Z\/p. The map c : H \u2192 Aut(H \u2229 Q) given by ch(x) = hxh\u22121 is well defined because Q is normal. Since |H| = pn and |Aut(H \u2229Q)| = p\u2212 1, we have that the map c is trivial. As a result we have that H < CG(Q). Assume now thatH\u2229Q = 0. In this caseH\u2229CG(Q) 6= H since otherwise we would have rk(G) > 3. Set K = H \u2229CG(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\u22121) 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 \u223c= (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 \u03b2 : G \u2192 C such that for any subgroup H \u2282 G, with H \u2229 Z(G) = 0, the restriction \u03b2|H is a complex character of H. If in addition H is a rank two elementary abelian subgroup, then the character \u03b2|H corresponds to an isomorphism class of fixed-point free representations. Proof. Define \u03b2 : G\u2192 C as follows: x 7\u2192 \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 (p2 \u2212 p)|G|, if x = 0; 0, if x \u2208 Z(G) \\ 0; \u2212p|G|, if x \u2208 Q \\ Z(G); 0, if x \u2208 CG(Q) \\Q; \u2212|G|, if x \u2208 G \\ CG(Q) of order p; 0 if x \u2208 G \\ CG(Q) of order greater than p. The map \u03b2 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 \u03c6k : Z\/pn \u2192 C; x 7\u2192 e2piikx\/n while \u03c6k\u03c6j : Z\/pn \u00d7 Z\/pn \u2192 C; (x, y) 7\u2192 e2pii(kx+jy)\/n. To lighten the notation, we drop the dependence in n on the \u03c6k\u2019s 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 \u2229Q 6= 0, then H \u2229Q \u223c= Z\/p and H \u223c= (H \u2229Q)\u00d7 Z\/p so that: \u03b2|H = |G| p\u22121\u2211 k=0 p\u22121\u2211 j=1 \u03c6k\u03c6j 24 2.3. Some p-groups If H \u2229Q = 0, then Z\/p \u223c= H \u2229CG(Q) so that H \u223c= (H \u2229CG(Q))\u00d7Z\/p and: \u03b2|H = (p\u2212 1)|G|\/p \uf8eb\uf8edp\u22121\u2211 k=0 p\u22121\u2211 j=1 \u03c6j\u03c6k \uf8f6\uf8f8+ |G| p\u22121\u2211 k=1 \u03c60\u03c6k. Consider now a subgroup H of G with H \u2229Z(G) = 0. We will proceed case by case using the classification above. 1. If H \u2229Q 6= 0 then H \u2282 CG(Q) and |K| = p with K = Q \u2229H. Let \u03c6 be the character of K which is p \u2212 1 on the identity and \u22121 for each other element of K. Then: \u03b2|H = p|G||H : K|Ind H K\u03c6. 2. If H\u2229Q = 0 and H \u2282 CG(Q) then H is cyclic and \u03b2|H = |G|\/|H|(p2\u2212 p)\u03c6 where \u03c6 is the character of H that is |H| on the identity and 0 elsewhere. 3. If H \u2229Q = 0 and H is cyclic with H \u2229 CG(Q) = 0, then |H| = p and \u03b2|H = |G|\/p\u03c6 where \u03c6 is (p3 \u2212 p2) on the identity and \u2212p elsewhere. 4. If H \u2229 Q = 0 and H is cyclic with H \u2229 CG(Q) 6= 0, then \u03b2|H = (p2 \u2212 p)|G|\/|H|\u2211|H|k=1 \u03c6k. 5. Assume that H \u2229 Q = 0 and that H is abelian of type (p, pn\u22121). Write H =< x, y \u2208 H|xp = ypn\u22121 = 1, [x, y] = 1 >. Notice that < y >= H \u2229CG(Q). For each 1 \u2264 i \u2264 p set Hi =< xyipn\u22122 >. Clearly |Hi| = p, Hi \u2229 CG(Q) = 0 and Hi \u2229Hj = 0 if i 6= j. Let \u03c6i be the character of Hi which is p \u2212 1 on the identity and \u22121 elsewhere. Set: \u03c6 = p\u2211 i=1 IndHHi\u03c6i. Since \u03c6(1) = |H|(p\u22121), \u03c6(z) = \u2212|H|\/p for z \u2208 H\\ < y > and \u03c6(z) = 0 for z \u2208< y >; we conclude that \u03b2|H = p|G|\/|H|\u03c6. 25 2.3. Some p-groups 6. If H \u2229Q = 0 and H \u223c= M(pn), we can write H =< x, y|xpn\u22121 = yp = 1, y\u22121xy = x1+pn+2 >. Let N =< xpn\u22122 , y >\u223c= (Z\/p)2 which is normal in H. Let \u03c6 be the character of N given by: \u03b2|H = (p\u2212 1) \uf8eb\uf8edp\u22121\u2211 i=0 p\u22121\u2211 j=1 \u03c6j\u03c6i \uf8f6\uf8f8+ p p\u22121\u2211 i=1 \u03c60\u03c6i. Then we have \u03c6(0) = p2(p\u22121), \u03c6((xkpn\u22122 , 0)) = 0, while \u03c6((xkpn\u22122 , yl)) = \u2212p. Finally \u03b2|H = IndHN |G|\/p|H : N |\u03c6. 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 \u00d7 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\u00d7Z\/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 \u223c= Sm \u00d7 Sn \u00d7 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 \u00d7 ...\u00d7Snk and if p > 3dim(X), then r \u2264 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\u22121xy = xm\u22121 > (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 G\u03c3 abelian for all cells \u03c3 \u2282 X. Then there is a free finite dimensional G-CW-complex Y ' X \u00d7 Sn1 \u00d7 ...\u00d7 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 G\u03c3 a representation \u03c1\u03c3 : G\u03c3 \u2192 U(m) such that \u03c1\u03c3|G\u03c4 \u223c= \u03c1\u03c4 whenever G\u03c4 < G\u03c3 and such that \u03c1\u03c3 is fixed point free for all G\u03c3 with rk(G\u03c3) = rkX(G). Consider the class function \u03b2 : G\u2192 C given by: x 7\u2192 \uf8f1\uf8f4\uf8f2\uf8f4\uf8f3 |G|(prkX(G) \u2212 1), if x = 0; \u2212|G|, if o(x) = p; 0, otherwise. To simplify the notation write A = G\u03c3 for an isotropy subgroup (which is abelian by hypothesis). We need to prove that \u03b2|A is a character which is fixed point free whenever A \u223c= (Z\/p)rkX(G). Set Ap = {0}\u222a{x \u2208 A|o(x) = p}. 27 2.4. Infinite groups Since A is abelian we have Ap \/ A. Fix an injection f : Ap \u2192 (Z\/p)rkX(G). Write \u03c10 : (Z\/p)rkX(G) \u2192 U(prkX(G) \u2212 1) for the reduced regular repre- sentation and let \u03c1 = \u03c10 \u25e6 f be the representation Ap \u2192 (Z\/p)rkX(G) \u2192 U(prkX(G) \u2212 1). Consider finally the representation of A given by \u03b7 = |G||Ap|\/|A|IndAAp\u03c1. Clearly \u03b7(0) = |G|(prkX(G)\u22121), \u03b7(x) = \u2212|G| if x \u2208 Ap\\0 while \u03b7(x) = 0 if x \/\u2208 Ap. As a result \u03b2|A = \u03b7. Let now A \u223c= (Z\/p)rkX(G). Clearly \u03b2|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\u00d7 ...\u00d7Snrk(G). The result in particular holds for extraspecial p-groups. Proof. Let X = Sn1\u00d7 ...\u00d7Snrk(Z(G)) be the product of the G-spheres arising from suitable representations of the center. Clearly rkX(G) = rk(G) \u2212 rk(Z(G)) and G\u03c3 is abelian. The conclusion follows. Remark 2.3.9. For extraspecial p-groups, similar results have been obtained independently by U\u030bnlu\u030b and Yalc\u0327in [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 \u0393 with finite virtual cohomological dimension. Recall that, by definition, a group \u0393 has finite virtual cohomological dimension, if it has a finite in- dex subgroup \u0393\u2032 < \u0393 with finite cohomological dimension (that is to say: Hn(\u0393\u2032) = 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(\u0393) = cd(\u0393\u2032) 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 \u0393 there exists a finite dimensional \u0393-CW-complex E\u0393 28 2.4. Infinite groups with |\u0393x| <\u221e for all x \u2208 E\u0393. 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 \u0393 with rank at most two finite subgroups. The easiest examples to consider are amalgamated products \u0393 = G1 \u2217G0 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 < \u0393, there is \u03b3 \u2208 \u0393 such that \u03b3H\u03b3\u22121 < Gi for i = 1 or i = 2 (see [34]). In particular rk(\u0393) = max {rk(G1), rk(G2)}. The first attempt would be to find an effective \u0393-sphere, i.e. a \u0393-sphere with rank at most one isotropy subgroups. We exhibit now an amalgamation of two p-groups which doesn\u2019t have an effective \u0393-sphere. Theorem 2.4.1. For p an odd prime, there is an infinite group \u0393 with fi- nite virtual cohomological dimension, satisfying the two following properties: every finite subgroup G < \u0393 is a p-group with rk(G) \u2264 2 and for every finite dimensional \u0393-CW-complex X ' Sn there is at least one isotropy subgroup \u0393\u03c3 with rk(\u0393\u03c3) = 2. Proof. Let E and E\u2032 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\u00d7 3 matrices over Fp with 1 on the diagonal). Consider the amalgamated product \u0393 = E\u2032 \u2217Z\/pE given by Z\/p = Z(E) and an injective map f : Z\/p\u2192 E with f(Z\/p) \u2229 Z(E\u2032) = 1. Clearly rk(\u0393) = 2. Let \u0393 act on a finite dimensional CW-complex X ' Sn. Consider the restriction of this action to E and E\u2032. 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\u2032 must be realized by characters \u03c7E and \u03c7E\u2032 . Clearly the dimension functions of \u03c7E and \u03c7E\u2032 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 \u03b1, giving rise to an effective sphere, vanishes outside Z(E) while \u03b1(z) = m\u03b6p for all z \u2208 Z(E)\\{0} (here \u03b6p is a p-root of the unity). Thus, \u03c7E and \u03c7E\u2032 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\u2019s 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\u2297 ...\u2297Kr is a complex of projective Z [G]-modules with H\u2217(K) \u223c= H\u2217(Sn1 \u00d7 ...\u00d7 Snr) (see [7]). In corollary 2.4.6 we prove a similar result: for every group \u0393 with vcd(\u0393) <\u221e, there are rk(\u0393) finite dimensional Z [\u0393]-complexesC1, ...,Crk(\u0393) such that D = C\u2217(E\u0393) \u2297C1 \u2297 ... \u2297Crk(\u0393) is a complex of projective Z [\u0393]- modules with H\u2217(D) \u223c= H\u2217(Sn1 \u00d7 ...\u00d7 Snrk(\u0393)). As a result, the group \u0393 introduced in theorem 2.4.1 satisfies the alge- braic analogue of the rank conjecture but does not have an effective \u0393-sphere. The geometric problem of knowing whether or not \u0393 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 \u03b6 \u2208 Hn(G,R) \u223c= ExtnRG(R,R) \u223c= HomRG(\u2126\u0302nR,R), where \u2126\u0302nR is the nth kernel in a RG-projective resolution P of R. We choose a cocycle \u03b6\u0302 : \u2126\u0302nR\u2192 R representing \u03b6. By making P large enough we can assume that \u03b6\u0302 is 30 2.4. Infinite groups surjective. We denote L\u03b6 its kernel and form the pushout diagram: L\u03b6 = \/\/ \u000f\u000f L\u03b6 \u000f\u000f 0 \/\/ \u2126\u0302nR \/\/ \u03b6\u0302 \u000f\u000f Pn\u22121 \/\/ \u000f\u000f Pn\u22122 \/\/ \u000f\u000f ... \/\/ P0 \/\/ \u000f\u000f R \/\/ = \u000f\u000f 0 0 \/\/ R \/\/ Pn\u22121\/L\u03b6 \/\/ Pn\u22122 \/\/ ... \/\/ P0 \/\/ R \/\/ 0 We denote by C\u03b6 the chain complex: 0\u2192 Pn\u22121\/L\u03b6 \u2192 Pn\u22122 \u2192 ...\u2192 P0 \u2192 R\u2192 0 formed by truncating the bottom row of this diagram. Thus we have that H0(C\u03b6) = Hn\u22121(C\u03b6) = R while Hi(C\u03b6) = 0 if i 6= 0, n \u2212 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 \u03be1, ..., \u03ber \u2208 H\u2217(G,Z) such that, for all H < G with rk(H) \u2264 r, the complex Z [G\/H]\u2297 L\u03be1 \u2297 ...\u2297 L\u03ber 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\u2217(C1 \u2297 ... \u2297 Cr) = H\u2217(Sn1 \u00d7 ...\u00d7 Snr); with C1 \u2297 ...\u2297Cr a complex of Z [H]-projective modules for all H < G with rk(H) \u2264 r. Proof. Let \u03be1, ..., \u03ber \u2208 H\u2217(G,Z) be the classes given in proposition 2.4.3. Consider the chain complex C = C\u03be1\u2297 ...\u2297C\u03ber . Clearly H\u2217(C) = H\u2217(Sn1\u00d7 ... \u00d7 Snr). For the second part of the claim, observe that all the modules in C\u03bei are Z [G]-projective except the module Pni\u22121\/L\u03bei . Recall that the tensor product of any module with a projective module is projective, so that it remains to examine the module Pn1\u22121\/L\u03be1 \u2297 ... \u2297 Pnr\u22121\/L\u03ber . Let 31 2.4. Infinite groups H < G be such that rk(H) \u2264 r. Since Z [G\/H] \u2297 L\u03be1 \u2297 ... \u2297 L\u03ber is Z [G]- projective by proposition 2.4.3, we conclude that Z [G\/H] \u2297 Pn1\u22121\/L\u03be1 \u2297 ... \u2297 Pnr\u22121\/L\u03ber is Z [G]-projective as in 5.14.2 of [9]. It then easily follows that Pn1\u22121\/L\u03be1 \u2297 ...\u2297 Pnr\u22121\/L\u03ber 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 \u0393 with vcd(\u0393) < \u221e and rank r. Write \u0393\u2032 for a torsion-free normal subgroup of \u0393 with G = \u0393\/\u0393\u2032 finite. We apply corollary 2.4.4 to \u0393\/\u0393\u2032 with r = rk(\u0393). We recover a Z [\u0393]- complex C1 \u2297 ... \u2297Cr such that H\u2217(C1 \u2297 ... \u2297Cr) = H\u2217(Sn1 \u00d7 ... \u00d7 Snr); with C1\u2297 ...\u2297Cr a complex of Z [H]-projective modules for all finite H < \u0393. On the other hand, we have that the Z\u0393-complex C\u2217(E\u0393) is contractible. Therefore the complex D = C\u2217(E\u0393) \u2297C1 \u2297 ... \u2297Cr is such that H\u2217(D) = H\u2217(Sn1 \u00d7 ...\u00d7 Snr). Lemma 2.4.5. With the notation above, the complex D is Z [\u0393]-projective. Proof. The complex C\u2217(E\u0393) decomposes as a graded direct sum of permu- tation modules: C\u2217(E\u0393) = \u2295\u03c3Z [\u0393\/\u0393\u03c3] = \u2295\u03c3Z [\u0393] \u2297Z[\u0393\u03c3 ] Z. Here \u03c3 spans the cells of E\u0393\/\u0393 and the grading is given by the dimensions of the cells \u03c3. Consequently D = \u2295\u03c3(Z [\u0393] \u2297Z[\u0393\u03c3 ] C1 \u2297 ... \u2297Cr), so that we only need to prove that Z [\u0393]\u2297Z[\u0393\u03c3 ]C1 \u2297 ...\u2297Cr is Z [\u0393]-projective. Let Q\u03c3 be a graded Z [\u0393\u03c3]-module such that (C1\u2297...\u2297Cr)\u2295Q\u03c3 is Z [\u0393\u03c3]-free. We then have that (Z [\u0393]\u2297Z[\u0393\u03c3 ]C1\u2297...\u2297Cr)\u2295(Z [\u0393]\u2297Z[\u0393\u03c3 ]Q\u03c3) = Z [\u0393]\u2297Z[\u0393\u03c3 ]((C1\u2297...\u2297Cr)\u2295Q\u03c3) is Z [\u0393]-free. Consequently, the algebraic version of the rank conjecture holds for groups of finite virtual cohomological dimension: Corollary 2.4.6. For a group \u0393 with vcd(\u0393) <\u221e and rk(\u0393) = r, there exist a finite dimensional contractible complex C\u2217(E\u0393) and r finite dimensional Z [\u0393]-complexes C1,...,Cr such that D = C\u2217(E\u0393)\u2297C1 \u2297 ...\u2297Cr is a Z [\u0393]- projective complex with H\u2217(D) \u223c= H\u2217(Sn1 \u00d7 ...\u00d7 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 \u2208 F} the set morphisms between two G orbits is given by: Mor(G\/H,G\/K) = {g \u2208 G|Hg < K} \/K. In what follows, we will write \u0393 = OrFG for the full orbit category and \u0393p 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 \u0393p is a full subcategory of \u0393. Let R be a commutative ring with unit. An R\u0393-module M is a contravariant functor: M : \u0393op \u2192 R\u2212Mod from the orbit category to the category of R-modules. A morphism of R\u0393- modules is then a natural transformation of functors. The category of R\u0393 modules is denoted by R\u0393\u2212Mod. Since \u0393 is a small category and R\u2212Mod is abelian, the category of R\u0393-modules is abelian as well. We are able, therefore, to do homological algebra in the category R\u0393\u2212Mod. For R\u0393- modules, the terms exact, injective, surjective, etc. are determined object wise. For instance, the sequence of R\u0393-modules: L\u2192M \u2192 N is exact if and only if the sequence of R-modules: L(G\/H)\u2192M(G\/H)\u2192 N(G\/H) 35 3.1. Homological algebra over the orbit category is exact for all G\/H \u2208 Ob\u0393. As usual, a R\u0393-module P is projective if and only if the functor: HomR\u0393(P,\u2212) : R\u0393\u2212Mod\u2192 R\u2212Mod is exact. For an orbit G\/H \u2208 Ob\u0393, define the free module generated at G\/H by: FG\/H(G\/K) = RMor(G\/K,G\/H) for all G\/K \u2208 Ob\u0393. Here RMor(G\/K,G\/H) is the free R-module on the set of R\u0393 morphisms from G\/K to G\/H. The R\u0393-module FG\/H is defined in such a way that maps of R\u0393-modules FG\/H \u2192M are determined by the image of IdG\/H \u2208 FG\/H(G\/H) in M(G\/H). In particular: HomR\u0393(FG\/H ,M) \u223c=M(G\/H) which says that FG\/H is a projective R\u0393-module for each G\/H \u2208 Ob\u0393. We are now able to give a definition of a free R\u0393-module: Definition 3.1.1. An R\u0393-module M is called free if it is isomorphic to:\u2295 G\/H\u2208\u03bb FG\/H for some collection of orbits \u03bb \u2282 Ob\u0393. Just as in the case of modules over a ring, one can prove that a R\u0393- module is projective if and only if it is a summand of a free module. Simi- larly, we say that a R\u0393-module M is finitely generated if there is a finitely generated free R\u0393-module covering it: \u2295ki=1FG\/Hi \u0010 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 Q\u0393-chain complex C is quasi-isomorphic to its homology H(C). 36 3.1. Homological algebra over the orbit category Proof. A Q\u0393-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\u2192 ...\u2192 Dn = Ker(dn)\u2295Ker\u22a5(dn)\u2192 ...\u2192 0 The projection map D \u2192 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 Q\u0393- 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)\u2192 Cn(G\/L) respect the scalar products. This is what we are going to show next. Since C is Q\u0393-free, all the Q-modules involved must necessarily be of the form QMor(G\/H \u2032, G\/H), for some H \u2032,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) \u2192 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) \u2192 Mor(G\/K,G\/H); u 7\u2192 ug for g \u2208 WK = NK\/K, the Weil group. To study Cn(G\/K) \u2192 Cn(G\/L), notice that all maps QMor(G\/K,G\/H)\u2192 QMor(G\/L,G\/H) respect the canonical scalar products because they are given by set injections g : Mor(G\/K,G\/H) \u2192 Mor(G\/L,G\/H); u 7\u2192 ug for elements g \u2208 Mor(G\/L,G\/K). (Each mor- phism g \u2208 Mor(G\/L,G\/K) being surjective makes the corresponding map g :Mor(G\/K,G\/H)\u2192Mor(G\/L,G\/H) injective). To summarize: we showed that for a perfect free Q\u0393-complex C, there are Q\u0393-modules Kern and Ker\u22a5n such that Cn = Kern \u2295Ker\u22a5n . Moreover, Kern(G\/H) = Ker(Cn(G\/H) \u2192 Cn\u22121(G\/H)). As a consequence, we can define a map of Q\u0393-complexes C \u2192 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 R\u0393-complexes (see [29]). Let K0(R\u0393) be the Groethendieck group of finitely generated pro- jective R\u0393-modules. Let F be subgroup of K0(R\u0393) generated by free R\u0393- modules. The reduced K-theory group, is the quotient K\u03030(R\u0393) = K0(R\u0393)\/F . For a perfect R\u0393-complex P , the sum o(P ) = \u2211\u221e i=1(\u22121)iPi defines a class o\u0303(P ) \u2208 K\u03030(R\u0393). If P is a perfect complex and if f : P \u2192 C is a quasi-isomorphism, then we set o\u0303(C) := o\u0303(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, o\u0303(C) = 0 if and only if C is weakly equivalent to a perfect free R\u0393-complex. If C is perfect, then o\u0303(C) = 0 if and only if C is homotopy equivalent to a perfect free R\u0393-complex. We would like to be able to express the obstruction theory for R\u0393- modules in terms of the obstruction theories of R [WH]-modules, forH some subgroups of G. The trouble here is that the functor ResH : [R\u0393\u2212Mod]\u2192 [R [WH]\u2212Mod]; M 7\u2192M(G\/H) does not take projective to projective! We need a better functor: For a R\u0393-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) \u2192 M(G\/H) induced by all non isomor- phisms f : G\/H \u2192 G\/K. Observe that M(G\/H)s is directly a R [WH]- submodule because for f \u2208WH, the composition g \u25e6 f is an isomorphism if and only if g is. Finally, we define the splitting functor SH : [R\u0393\u2212Mod]\u2192 [R [WH]\u2212Mod] by M 7\u2192 M(G\/H)\/M(G\/H)s. As in [29], one can then prove that: Proposition 3.1.4. The splitting functors SH respect \u201ddirect sum\u201d, \u201dfinitely generated\u201d, \u201dfree\u201d and \u201dprojective\u201d. To compute the obstruction class, we now have theorem 10.34 of [29]: Theorem 3.1.5. The reduced K-theory splits as: K\u03030(R\u0393) \u223c= \u2295(H). We order the category \u0393 as follows: (G\/G,G\/C3, G\/C2, G\/e). We want to find chain complexes D3 \u2208 Z(3)\u0393 and D2 \u2208 Z(2)\u0393 realizing the dimension function (\u22121,\u22121, 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\u2217(\u03a33,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)\u03a33-module for i = 0, ..., 3. Since the free module generated at G\/e is FG\/e = (0, 0, 0,Z(3)\u03a33), we recover the following free Z(3)\u0393-complex D\u20323: [3] M3 f3 \u000f\u000f [2] M2 f2 \u000f\u000f [1] M1 f1 \u000f\u000f [0] M0 \u03a33\/\u03a33 \u03a33\/C3 \u03a33\/C2 \u03a33\/e Its dimension function is (\u22121,\u22121,\u22121, 3). Not quite what we want. Observe first that the map of G-sets pi : G\u2192 G\/C2 \u223c= C3 induces an exact sequence of Z(3)\u03a33-modules: 0 \/\/ Ker(pi) \/\/ Z(3)\u03a33 pi \/\/ Z(3)C3 \/\/ 0 . The map \u00b5 : Z(3)C3 \u2192 Z(3)\u03a33, given by [ (\u03c3i, \u03c4 j) ] 7\u2192 (\u03c3i, \u03c40) 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)\u03a33-splitting of the exact sequence, by setting \u00b5G : Z(3)C3 \u2192 Z(3)\u03a33, [ (\u03c3i, \u03c4 j) ] 7\u2192 ((\u03c3i, \u03c40) + (\u03c3i, \u03c41))\/2. In particular, the free 5C3 is the subgroup of \u03a33 generated by \u03c3, while C3 is the \u03a33-set given by \u03a33\/C2 40 3.2. Finite homotopy G-spheres module generated at G\/e splits as FG\/e = (0, 0, 0,Z(3)C3 \u2295Ker(pi)). Recall also that the free Z(3)\u0393-module generated at G\/C2 is (0, 0,Z(3),Z(3)C3). We can now modify D\u20323 into a perfect free Z(3)\u0393-complex D3, which re- alizes the dimension function (\u22121,\u22121, 3, 3). To describe the new complex, we use the symbol \u03c1i to designate the projection of a direct sum into its i-th component. We leave out the columns \u03a33\/\u03a33 and \u03a33\/C3 because they are trivial: [4] Z(3)C3 \u2295Ker(pi) \u03c11\u22950\u2295\u03c12\u22950 \u000f\u000f [3] Z(3) 0 \u000f\u000f Z(3)C3 \u2295 Z(3)C3 \u2295Ker(pi)\u2295M3 0\u2295\u03c12\u22950\u2295(f3\u25e6\u03c14) \u000f\u000f [2] Z(3) 1 \u000f\u000f Z(3)C3 \u2295 Z(3)C3 \u2295Ker(pi)\u2295M2 \u03c11\u22950\u2295\u03c13\u2295(f2\u25e6\u03c14) \u000f\u000f [1] Z(3) 0 \u000f\u000f Z(3)C3 \u2295 Z(3)C3 \u2295Ker(pi)\u2295M1 \u03c12\u2295(f1\u25e6\u03c14) \u000f\u000f [0] Z(3) Z(3)C3 \u2295M0 \u03a33\/C2 \u03a33\/e Next, we construct a free Z(2)\u0393-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)\u0393-complex D\u20322: [2] Z(2)[\u03a33] f2 \u000f\u000f [1] Z(2) 0 \u000f\u000f Z(2)C3 \u2295 Z(2)[\u03a33] f1\u25e6\u03c12 \u000f\u000f [0] Z(2) Z(2)C3 \u03a33\/\u03a33 \u03a33\/C3 \u03a33\/C2 \u03a33\/e The maps f1 and f2 are given by: f1(e) = \u03c3 \u2212 \u03c32 f2(e) = (\u03c3, e\u2212 \u03c3\u03c4 \u2212 \u03c32\u03c4). Over the orbits G\/\u03a33, G\/C3 and G\/C2, everything is clear. We need to study the Z(2)\u03a33-complex sitting over the orbit G\/e. It is easy the check that Im(f1 \u25e6 \u03c12) = Ker(\u000f : Z(2)[\u03a33\/C2] \u2192 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)\u03a33 \u2192 Ker(f1 \u25e6 \u03c12) is given by:\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed 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 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 Since 9 is invertible in Z(2), we have that Ker(0 + f1)\/Im(f2) \u223c= Z(2) and Ker(f2) = 0. Finally, by concatenating D\u20322 with itself, we obtain the desired 42 3.2. Finite homotopy G-spheres perfect free Z(2)\u0393-complex D2 realizing (\u22121,\u22121, 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\u2019s 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 R\u0393-complexes: Given a projective R\u0393 chain complex C, there is a sequence of projective chain complexes C(i) together with maps fi : C \u2192 C(i) inducing homology isomorphisms for dimension \u2264 i. Moreover, there is a tower of maps: C(i) \u000f\u000f C(i\u2212 1) \u000f\u000f\u001f \u001f \u001f \u03b1i \/\/ \u03a3i+1P (Hi) C EE ;;xxxxxxxxxx \/\/ ##F FF FF FF FF F C(1) \u000f\u000f \u03b12 \/\/ \u03a33P (H2) C(0) \u03b11 \/\/ \u03a32P (H1) such that C(i) = \u03a3\u22121C(\u03b1i), where C(\u03b1i) denotes the algebraic mapping cone of \u03b1i 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 \u2192 D is defined as the chain complex C(f) = D\u2295\u03a3C with boundary map \u2202(x, y) = (\u2202(x) + f(y), \u2202y). Note that \u03a3n is the shift operator for chain complexes given by (\u03a3nC)i = Ci\u2212n. 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 ExtiR\u0393(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 \u2192 C(d) is a homology isomorphism, so that C and C(d) are homo- topy equivalent. As a consequence, ExtiR\u0393(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 \u2208 P we are given a perfect free Z(pi)\u0393-complex Dpi such that H(Dpi\u2297Q) is isomorphic to H(Dpj\u2297Q) for all i and j. If there is a perfect free Z(1\/P )\u0393-complex D1\/P such that H(D1\/P \u2297Q) is isomorphic to H(Dpj \u2297 Q), then there is a perfect free Z\u0393-complex D with D \u2297 Z(pi) quasi-isomorphic to Dpi for all pi \u2208 P and D \u2297 Z(1\/P ) quasi-isomorphic to D1\/P . Proof. Consider the limit: D = lim uukkkk kkkk kkkk kkkk k yysss sss sss sss \u000f\u000f %%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 \u000f\u000f ... yyttt tt tt tt tt Dpn uukkkk kkk kkk kkk kkk H(Dp1 \u2297Q) Observe that the maps into H(Dp1\u2297Q) 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\u2192 D \u2192 Dp \u2295D1\/p \u2192 H(Dp)\u2297Q\u2192 0 which in turn yields a long exact sequence: ...\u2192 Hn(D)\u2192 Hn(Dp \u2295D1\/p)\u2192 Hn(Dp)\u2297Q\u2192 ... Applying the exact functor \u2212\u2297 Z(p), gives an exact sequence: ...\u2192 Hn(D)\u2297Z(p) \u2192 (Hn(Dp)\u2295Hn(D1\/p))\u2297Z(p) \u2192 Hn(Dp)\u2297Q\u2297Z(p) \u2192 ... Since Q\u2297Z(p) = Q = Z(1\/p)\u2297Z(p), the map Hn(D1\/p)\u2297Z(p) \u2192 Hn(Dp)\u2297Q is surjective for all n, hence: Hn(D)\u2297 Z(p) \u223c= Ker [ (Hn(Dp)\u2295Hn(D1\/p))\u2297 Z(p) \u2192 Hn(Dp)\u2297Q ] Such a kernel is nothing but the pullback: Hn(Dp) \/\/ \u000f\u000f Hn(Dp) \u000f\u000f Hn(D1\/p)\u2297Q \u223c= \/\/ Hn(Dp)\u2297Q We readily conclude that Hn(D)\u2297Z(p) \u223c= Hn(Dp). Similarly Hn(D)\u2297Z(q) \u223c= Hn(D1\/P )\u2297 Z(q) for all q \/\u2208 P , so that Hn(D)\u2297 Z(1\/P ) \u223c= 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 Z\u0393- 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 \u2297 Z(pi) and M \u2297 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 ExtkZ\u0393(D,M) for all Z\u0393-modules M . As Z(pi) is flat over Z, we have that Ext k Z\u0393(D,M)\u2297 Z(pi) = Ext k Z(pi)\u0393 (D \u2297 Z(pi),M \u2297 Z(pi)) = ExtkZ(pi)\u0393(Dpi ,M \u2297 Z(pi)). By assumption, Dpi is perfect so that Ext k Z(pi)\u0393 (Dpi ,M \u2297 Z(pi)) = 0 for all k > dim(Dpi) and for all pi \u2208 P . Similarly, we have that ExtkZ\u0393(D,M) \u2297 Z(1\/P ) = ExtkZ(1\/P )\u0393(D1\/P ,M \u2297 Z(1\/P )) = 0 for all k > dim(D1\/P ) As a result, ExtkZ\u0393(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 Z\u0393-complex, we can talk about its obstruction o\u0303(D) \u2208 K\u03030(Z\u0393). Recall that o\u0303(D)\u2297Z(pi) = o\u0303(D \u2297 Z(pi)) = o\u0303(Cpi) for all pi \u2208 P and of course o\u0303(D) \u2297 Z(1\/P ) = o\u0303(D \u2297 Z(1\/P )) = o\u0303(D1\/P ). This implies that D is quasi-isomorphic to a projective free Z\u0393-complex, provided that all the complexes Dpi and D1\/P are free. Now that we can build perfect free Z\u0393-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)\u2192 N monotone if n(G\/K) \u2264 n(G\/H) whenever (H) < (K). An ROrFG-complex C is called an n-Moore complex, if it is connected and if H\u0303i(C(G\/H)) = 0 for i 6= n(G\/H). Theorem 3.2.5. [22] Let \u0393 = OrFG and consider a perfect free Z\u0393-complex C. Suppose that C is an n-Moore complex with n(G\/H) \u2265 3 for all G\/H \u2208 Ob(\u0393). Suppose further that Ci(G\/H) = 0 for all i \u2265 n(G\/H) + 1 and for all G\/H \u2208 Ob(\u0393). Then there is a finite G-CW-complex X such that C(X) is homotopy equivalent to C as a Z\u0393-complex. Moreover Gx \u2208 F for all x \u2208 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 \u2264 r and for all j \u2265 3, there is a finite homotopy G-sphere X with: DimqX(G\/Cqt) = { j + 2qr if t \u2264 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 \u2264 r there is a perfect free Z(p)\u0393-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)\u0393-functor generated at G\/Cqs has value: FG\/Cqs (G\/K) = \uf8f1\uf8f4\uf8f2\uf8f4\uf8f3 Z(p)Cqr\u2212s if e 6= (K) < (Cqs), Z(p)Cp o Cqr\u2212s if e = K, 0 otherwise. Consider the map FG\/Cqs \u2192 FG\/Cqs induced by the group ring norm map N : Z(p)Cqr\u2212s \u2192 Z(p)Cqr\u2212s , t 7\u2192 t\u2212 t2; where t is a generator of Cqr\u2212s . This gives rise to a Z(p)\u0393-complex T \u2032p with: Hi(T \u2032p(G\/K)) = \uf8f1\uf8f4\uf8f2\uf8f4\uf8f3 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 \u2032) = 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 \u223c= Z(p)Cp\u2295M , as Z(p)G-modules. Following the construction of example 3.2.2 again, one can easily define a perfect free Z(p)\u0393-complex Tp: Z(p)Cp \u2295M \/\/ Z(p)Cp \u2295M \u2295 FG\/Cqs \/\/ FG\/Cqs with the required homology. Lemma 3.2.8. The Z(q)G-module Ker(\u000f : Z(q)Cp \u2192 Z(q)) is projective. Proof. The short exact sequence of coefficients: 0 \/\/ Ker(\u000f) \/\/ Z(q)Cp \u000f \/\/ Z(q) \/\/ 0 induces a long exact sequence in homology: ... \/\/ Hn(G,Ker(\u000f)) \/\/ Hn(G,Z(q)Cp) \u223c= \/\/ 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(\u000f)) = 0, so that Ker(\u000f) 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 \u2208 Z(p)\u0393 and Dq \u2208 Z(q)\u0393 with the required homologies. Con- sider first the prime p. By [38], there is a chain complex D\u2032p: 0 \/\/ Z(p) \/\/M2qr\u22121 f2qr\u22121 \/\/ ... f1 \/\/M0 \/\/ Z(p) \/\/ 0 withMi a perfect free Z(p)G-module for all i = 0, ..., 2qr\u22121. We can look at this chain complex as a perfect free Z(p)\u0393-complex, with zeroes away from the orbit G\/e. Let T \u22172q 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 \u22172q r p \u2295D\u2032p 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 \u2192 FG\/Cqs induced by the norm map N : Z(q)Cqr\u2212s \u2192 Z(q)Cqr\u2212s . As in lemma 3.2.7, it defines a chain complex L, with homology: Hi(L(G\/K)) = \uf8f1\uf8f4\uf8f2\uf8f4\uf8f3 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(\u000f : Z(q)Cp \u2192 Z(q)) is summand of some free Z(q)G-module F = Q \u2295 B. We can then easily produce a chain complex D\u2032q of the form F \u2192 F \u2295 FG\/Cqs \u2192 FG\/Cqs with homology: Hi(D\u2032q(G\/K)) = { Z(p) if (K) < (Cqs), 0 otherwise. for i = 0, 1 andHi(D\u2032q) = 0 for i > 1 The required perfect free Z(q)\u0393-complex Dq is found again by concatenating (D\u2032) \u22172qr 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)\u0393 complex D1\/pq realizing the given dimension function. We conclude that 3.2.4 provides a perfect free Z\u0393-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(\u2212) } p||G| = { DimpY (\u2212) } 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\u00d7Sn, Ann. Math. 66, (1957). [16] F. Connolly, S. Prassidis. Groups which acts freely on Rm \u00d7 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. Yalc\u0327in. 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 \u00d7 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\u2019IHES 56, (1982), 129-169. [43] O. U\u030bnlu\u030b. Constructions of free group actions on products of spheres, Phd Thesis, University of Wisconsin, Madison, (2004). [44] O. U\u030bnlu\u030b, E Yalc\u0327in. Fusion systems and constructing free actions on products of spheres, arxiv. 54","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"2011-11","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0080542","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Mathematics","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"Attribution-NonCommercial-NoDerivatives 4.0 International","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#rightsURI":[{"value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Graduate","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/title":[{"value":"Group actions on homotopy spheres","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/type":[{"value":"Text","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#identifierURI":[{"value":"http:\/\/hdl.handle.net\/2429\/35981","type":"literal","lang":"en"}]}}