Homotopy Theoretic Methods in Group Cohomology

107 Pages·2001·6.199 MB·English

by  William G. Dwyer, Hans-Werner Henn

Advanced Courses in Mathematics CRM Barcelona
William G. Dwyer
Hans-Wcrncr Henn
Homotopy Theoretic Methods
in Group Cohomology
CtNtttCt ttCIRCAMAlUMTtCA
Birkhauser

William G. Dwyer
Hans-Werner Henn
Homotopy Theoretic Methods
in Group Cohomology
Birkhauser Verlag
Basel Boston Berlin
(cid:127) (cid:127)
Authors'addresses:
WilliamG.Dwyer Hans-WernerHenn
DepartmentofMathematics DepartementdeMathematiques
UniversityofNotreDame UFRdeMathematiquesetInformatique
NotreDame,IN46556 7rueReneDescartes
USA 67084StrasbourgCedex
France
e-mail:[email protected] e-mail:[email protected]
2000MathematicalSubjectClassification20J05;20J06,14F35,55U10
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Homotopytheoreticmethodsingroupcohomology/WilliamG.Dwyer;Hans-Werner
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Dedicated to
, ,
Chuck Chris Sarah
Jeanie and Davie
Contents
Preface ix
Classifying Spaces and Homology Decompositions
William G. Dwyer
1 Introduction 1
2 Classifyingspaces 2
3 Simplicial complexes and simplicialsets 6
4 Simplicial spaces and homotopy colimits 16
5 Nervesof categories and the Grothendieck construction 22
6 Homotopy orbit spaces 26
7 Homologydecompositions 28
8 Sharp homology decompositions. Examples 33
9 Reinterpreting the homotopycolimit spectralsequence 36
10 Bredon homology and the transfer 40
11 Acyclicity for G-spaces 43
12 Non-identity p-subgroups 45
-
13 Elementary abelian psubgroups 46
14 Appendix 49
References 51
viii Contents
Cohomology of Groups and Unstable Modules
over the Steenrod Algebra
Hans-Werner Henn
0 Introduction 55
1 Cohomology of groups 57
1.1 Some fundamental structural results for cohomologygroups
and cohomology ringsof finite groups 57
1.2 Quillen’s F-isomorphism theorem 58
2 Quillen’s F-isomorphism theorem from
the point of viewof unstable modules 63
2.1 Unstable modulesover the Steenrod algebra 63
2.2 Quillen’s F-isomorphism theorem from the point of view
of unstable modules 66
3 Lannes’T-functor and Lannes’generalization
of Quillen’s theorem 68
3.1 Review of Lannes’T-functor 68
3.2 Lannes’ generalizationof Quillen’stheorem 71
3.3 An outlineof a proof of Quillen’s Theorem 74
4 Approximations up to higher nilpotency 75
4.1 Localizations in abeliancategories 75
-
4.2 Localization away from n nilpotent modules 77
-
5 Commutative algebraof unstablefFBG modules;
approximationsof H*DG upto torsion modules 83
6 The centralizer spectral sequence 90
References 96
Preface
This book consists essentially of notes which were written for an Advanced
Course on Classifying Spaces and Cohomology of Groups. The course took
placeattheCentredeRecercaMathematica (CRM)inBellaterrafromMay27
toJune 2,1998and was part of an emphasissemester on AlgebraicTopology.
It consisted of two parallel series of 6 lectures of 90 minutes each and was
intended as an introduction to new homotopy theoretic methods in group
cohomology.
The first part of the book is concerned with methods of decomposing
the classifyingspace of a finite group into pieces made of classifying spacesof
appropriatesubgroups.Suchdecompositionshavebeenusedwithgreatsuccess
inthelast10-15yearsinthehomotopytheoryofclassifyingspacesofcompact
Lie groups and p-compact groups in the sense of Dwyer and Wilkerson. For
simplicity theemphasis hereisonfinitegroupsand on homological properties
. .
ofvariousdecompositionsknownascentralizerresp normalizerresp subgroup
decomposition. A unified treatment of the various decompositions is given
and the relations between them areexplored.This is proceeded by a detailed
discussionof basic notionssuchasclassifyingspaces,simplicialcomplexesand
.
homotopy colimits
A second crucial ingredient for the progress in the homotopy theory of
classifyingspacesof compact Liegroupsandp-compact groupscamefrom the
theoryof unstablemodulesover theSteenrodalgebra,in particularthetheory
of Lannes’ functor Ty.The second part discusses how this theory advances
our understanding of the mod-p cohomology ring H*BG of (a suitable class
of) groupsG and leadsin favorablecaseseven tocomplete calculations. After
recalling some classical results in cohomology of groups, unstable modules
sure introduced, Lannes’ theory is outlined and his calculation of TVH*BG
explained. Quillen’s theory of F-isomorphisms is shown to be equivalent to
Lannes’calculationofTyH*BGindegree0.Amajorthemeofthissecondpart
is then toexploit the informationgiven by the full computation ofTyH*BG.
Furthermore it is shown how partial knowledge of TyH*BG can be used to
give various approximationsto H*BG.
There is some overlap between the two parts in so far as centralizer
decompositions arise naturally in the approximations to H*BG discussed in
the last two sections of part II. Nevertheless the two parts remain logically
.
independant
We would like to thank the participants of the course for being such a
stimulatingaudience.OurthanksgoalsotothestaffoftheCRMandthetopol-
ogists at the Universitat Autonoma de Barcelona and in particular to Carles
Broto for organizing thiscourseand makingit such a pleasant experience.
January 2001 William G. Dwyer, Hans-Werner Henn
Classifying Spaces and
Homology Decompositions
W.G. Dwyer
Abstract. Suppose that G is a finite group. We look at the problem of
expressing the classifyingspace BG, up to mod p cohomology, as a ho-
motopy colimit of classifying spaces of smaller groups. A number of in-
teresting toolscome intoplay,such assimplicialsets and spaces, nerves
of categories,equivariant homotopy theory, and thetransfer.
1. Introduction
In these notes we discuss a particular technique for trying to understand the
classifyingspace BGof afinitegroupG.The techniqueisespecially usefulfor
studyingthecohomology of BG, but it canalsoserveother purposes.Theap-
proach is pick a prime number pand construct BG, up to mod pcohomology,
bygluingtogether classifyingspacesof propersubgroupsof G. Aconstruction
like this is called a homology decomposition of BG; in principle it gives an
inductive waytoobtain informationabout BG frominformation about classi-
fying spaces of smaller groups. These homology decompositions are certainly
interesting on their own, but another reason to work with them is that it il-
lustrates how to use some everyday topological machinery. We try to make
themachineryeasiertounderstandbyexplainingsomethingsthat areusually
taken for granted.
.
The outline of the paper is this Section 2 introduces classifying spaces
.
and shows how to construct them Section 3faces the issue that for our pur-
posesitismucheasiertoworkwithcombinatorialmodelsfortopologicalspaces
than withtopologicalspacesthemselves.Thereisanextended attempt tomo-
tivate the particular combinatorial models we will use, called simplicial sets;
the section ends with a description of the simplicial sets that correspond to
classifyingspaces.Section4givesasystematicaccountofalargeclassofgluing
constructionscalled homotopy colimits.Theeasiest way todescribehomotopy
colimits is with the helpof simplicial spaces (although fromthe point of view
ReceivedbytheeditorsJune13, 2001.