Equivariant homology theories for totally disconnected groups [Elektronische Ressource] / vorgelegt von Juliane Sauer, geb. Jänich

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Juliane SauerEquivariant Homology Theoriesfor Totally Disconnected Groups2002Reine MathematikEquivariant Homology Theoriesfor Totally Disconnected GroupsInaugural-Dissertationzur Erlangung des Doktorgradesder Naturwissenschaften im Fachbereich Mathematikder Mathematisch-Naturwissenschaftlichen Fakult atder Westf alischen Wilhelms-Universit at Munstervorgelegt vonJuliane Sauer, geb. J anichaus Regensburg2002Dekan: Prof. Dr. F. NattererErster Gutachter: Prof. Dr. W. Luc kZweiterhter: Prof. Dr. H. HammTag der mundlic hen Prufungen: 20.8./23.8./26.8.2002Tag der Promotion: 26.8.2002Equivariant Homology Theories for TotallyDisconnected GroupsJuliane SauerAbstractThe notion of an equivariant family of spectra corresponds to the notion of anequivariant homology theory as used in [Luc02]. A general principle how to con-struct equivariant families of spectra will be given. This machine can be used tode ne many interesting equivariant homology theories. The main examples will bealgebraic K- and L-theory for discrete groups and topological K-theory, HochschildHomology, Cyclic Homology and Periodic Homology for totally disconnected, lo-cally compact groups.In the appendix equivariant K-theory (cohomology) for proper actions of totallydisconnected groups will be considered. We will show that in general it cannot bede ned via equivariant vector bundles as for discrete groups, because of the failureof the excision axiom.

Sujets

• Mathematics

Informations

• Publié par: westfalische_wilhelms-universitat_munster
• Publié le: 01 janvier 2002
• Nombre de lectures: 13
• Langue: English

Extrait

Juliane Sauer

Equivariant Homology Theories for Totally Disconnected Groups

2002

Reine Mathematik

Equivariant Homology Theories for Totally Disconnected Groups

Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Mathematik derMathematisch-NaturwissenschaftlichenFakult¨at derWestf¨alischenWilhelms-Universita¨tM¨unster

vorgelegt von JulianeSauer,geb.Ja¨nich aus Regensburg 2002

Dekan: Erster Gutachter: Zweiter Gutachter: Tagdermu¨ndlichenPru¨fungen: Tag der Promotion:

Prof. Dr. F. Natterer Prof.Dr.W.L¨uck Prof. Dr. H. Hamm 20.8./23.8./26.8.2002 26.8.2002

for

Equivariant Homology Theories Disconnected Groups

Juliane Sauer

Abstract

Totally

The notion of an equivariant family of spectra corresponds to the notion of an equivarianthomologytheoryasusedin[L¨uc02].Ageneralprinciplehowtocon- struct equivariant families of spectra will be given. This machine can be used to define many interesting equivariant homology theories. The main examples will be algebraic K- and L-theory for discrete groups and topological K-theory, Hochschild Homology, Cyclic Homology and Periodic Homology for totally disconnected, lo- cally compact groups. In the appendix equivariant K-theory (cohomology) for proper actions of totally disconnected groups will be considered. We will show that in general it cannot be defined via equivariant vector bundles as for discrete groups, because of the failure of the excision axiom.

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Hochschild, Cyclic and Periodic Homology 5.1 Simplicial Objects . . . . . . . . . . . . . . 5.2 Simplicial Spectra and Quasi-Fibrations . . 5.3 Hochschild Homology, Cyclic Homology and egories with Cofibrations . . . . . . . . . . . 5.4 A Spectrum for Hochschild Homology . . . 5.5 A Spectrum for Cyclic Homology . . . . . . 5.6 A Spectrum for Periodic Homology . . . . . 5.7 The Long Exact Sequence ofH HandH C.

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. . . . . . . . . . . . Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Topological K-theory 6.1C ∗ . . . . . . . . . . . . . . . . . . . .-Categories . . . . . . . . . . . . . . 6.2 A Spectrum for Topological K-theory . . . . . . . . . . . . . . . . . . . . .

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Appendix: K-Theory for Proper Actions of a Totally Disconnected Group 8.1G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  .-vector bundles 8.2 Equivariant K-Theory for proper smoothG-CW-complexes: Definition and First Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Excision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Equivariant K-theory for proper actions of prodiscrete groups . . . . . . . 8.5 The Chern Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Induction

Maps for Spaces on which the Kernel doesn’t act freely

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Smooth Equivariant Families of Spectra 2.1 Smooth Equivariant Homology Theories . . 2.2 Proof of the Main Theorem . . . . . . . . .

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Topological Groupoids with Haar Systems 4.1 Haar Measures . . . . . . . . . . . . . . . . 4.2 The Hecke-Algebra . . . . . . . . . . . . . . 4.3 Topological Groupoids with Haar Systems .

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The Orbit Category and Spaces over 1.1 Spaces and Spectra over a Category 1.2 The Orbit Category . . . . . . . . . 1.3G-Homology Theories . . . . . . . . 1.4 Totally Disconnected Groups . . . . 1.5 Categories . . . . . . . . . . . . . . .

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the Orbit Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction

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Algebraic K- and L-theory

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Contents

Introduction

In[L¨uc02]Lu¨ckconstructsaCherncharacterforproperequivarianthomologytheo- ries. The main result is the following: LetH ∗ ? be a proper equivariant homology theory with values inR-modules, whereRis a commutative ring containing the rationals. Sup- pose for every discrete groupGthat theRSub(GF IN)-moduleH qG (G/?) is flat for all q≥0. Then there is a natural isomorphism ch ∗ G (X A) : M H p Or(G) (X A;H qG (G/?))→ H ∗ G (X A). p+q=∗

A(smooth) equivariant homology theoryH ∗ ? is an assignment which associates to every discrete (totally disconnected, locally compact) groupGa (smooth)G-homology theory H ∗ G together with the following so calledinduction structure: Letα:G→Mbe a (continuous open) group homomorphism, (X A) a (smooth)G-CW-pair on which ker(α for every) acts freely. Thenn∈Zthere is a natural isomorphism

G ind α :H n (X A)→ H nM (ind α (X A))

which is compatible with the boundary homomorphisms, functorial inαand satisfies a certain condition ifαis conjugation with a group element. TheCherncharacterin[L¨uc02]isexplicitlyconstructedwiththehelpoftheinduction homomorphisms ofH ? ∗ .

Our main theorem gives a general method how to construct (smooth) equivari- ant homology theories using Or(G for any discrete (totally discon-)-spectra. Namely nected, locally compact) groupGletG G be the Or(G)-groupoid (Or(GO)-groupoid) withOb(G G (G/H)) =G/Hand mor GG(G/H) (g 1 H g 2 H) =g 2 H g 1 −1 . Then the following theorem holds:

Theorem 2.10LetE:Gr→Spt(Top)(resp.E:TGr→Spt(Top)) be a functor with the property that it maps equivalences of (topological) groupoids to weak equivalences of spectra. Then mapsi α can be constructed such that(E ? :=E◦G ?  i ? )is a (smooth) equivariant family of spectra (and henceH ∗ ? (∙;E◦ G ? )a (smooth) equivariant homology theory).

HereH ∗ G (∙;E◦ G G ) is theG-homology theory with coefficients in the Or(G)- spectrumE◦G G . In Chapter 1 this and other basic notions concerning the orbit-category and spaces over the Orbit category will be explained. In Chapter 2 we will think about what could or should be an induction structure for equivariant spectra and how this can be mirrored on equivariant groupoids. Finding the right formulation for this is already half of the proof of the main theorem. In the second part of the paper applications for the main theorem are given, including K- and L-theory for discrete groups, topological K-theory, Hochschild Homology, Cyclic Homology and Periodic Homology for totally disconnected, locally compact groups. Applying the main theorem to get examples of smooth equivariant homology theo- ries, i.e. constructing a functorE:Gr→Spt(Top many In) often takes two steps. cases there exists or can be constructed an interesting functorF:Cat < →Spt(Top) from some subcategory of the category of small categories, for example the split exact categories, to the category of spectra.

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Then we have to choose a suitable functorGr→Cat < which assures that we get the right coefficients, namely the group ring for discrete groups and the Hecke algebra resp. the reduced groupC ∗ -algebra for totally disconnected groups. the case of K-theory In for example, this functor assigns to any groupoidGthe symmetric monoidal category P(ZG ⊕ ), whereZGis theZ-linear category associated toG,ZG ⊕ is the category of finite tuples of objects ofZG, andPassigns to any category its idempotent completion. ForK-andL-theoryweusetheconstructionofDavisandL¨uckin[DL98],for Hochschild Homology, Cyclic Homology and Periodic Homology we use work of Mc- Carthy, who defines in [McC94] Hochschild Homology, Cyclic Homology and Periodic Homology fork-linear categories with cofibrations,k def-a commutative ring. This inition leads to the construction of spectra for these homology theories. A functor K top :C ∗ Cat→Spt(Top In) is constructed in [Joa02]. both cases we give a detailed construction of the functorsTGr→Cat < mentioned above. For every totally disconnected, locally compact groupGand everyG-homology the- oryH ∗ G the projectionE(GCO)→ ∗, whereCOis the family of compact open sub- groups, gives rise to anassembly map

a:H ∗ G (E(GCO))→ H ∗ G (G/G).

The correspondingisomorphism conjecture Withsays that this map is an isomorphism. the above constructed spectra isomorphism conjectures for Hochschild Homology and Cyclic resp. Periodic Homology can be formulated. In Section 7 we show that the construction of the main theorem can often be extended in the sense that induction homomorphisms can also be defined forG-CW-complexes on which the kernel of the group homomorphism in question doesn’t act freely.

The Appendix shows a different aspect of the action of totally disconnected groups. For proper actions of discrete groups, topological K-theory (cohomology) can be defined using equivariant vector-bundles (see [LO01]). We show that the analogous statement for proper smooth actions of totally disconnected groups isn’t true in general, an explicit counterexample will be given where excicion doesn’t hold. For groups which are an inverse limit of discrete groups, the definition carries through and the Chern character of [LO99] can be extended to these groups.

IwouldliketothankmyadvisorWolfgangL¨uckforhisencouragementandsupport. Also I would like to thank Holger Reich for some very helpful discussions and suggestions.

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1

1.1

The Orbit Category and Spaces over the Orbit Cat- egory

Spaces and Spectra over a Category

For an extensive treatment of spaces over a category see [DL98]. LetC Abe a small category.covariant (contravariant) space over the categoryCor C-spaceis a covariant (contravariant) functor

X:C →Top

whereTop map betweenis the category of compactly generated topological spaces. A C-spaces is a natural transformation of such functors. ApointedC-spaceis a functor fromCtoTop + and aweak equivalencebetween two C-spaces is a map ofC-spaces which at each object is a weak equivalence of topological spaces. AC-spectrum(resp.C-Ω-spectrum) is a functor fromCtoSpt(Top) (resp. ΩSpt(Top)).

Remark 1.1The objects of the categorySpt(Top) are thespectra, i.e. families{E n  n∈ Z}of pointed topological spaces together with structure mapss n :S 1 ∧E n →E n+1 . The morphisms are thestrict mapsof spectra, i.e. a morphismf:E→Fbetween two spectra consists of mapsf n :E n →F n which are compatible with the structure maps: f n+1 ◦s n E =s n F ◦(S 1 ∧f n ). A spectrum is an Ω-spectrumif the adjointst n :E n →ΩE n+1 of the structure maps are weak equivalences. The categoryΩSpt(Top) is the full sub- category ofSpt(Top) with objects the Ω-spectra. The(stable) homotopy groupsπ ∗ (E) of a spectrumEare defined to be

π n (E) := l − i → mπ n+k (E k ) n∈Z k∈N

where the structure maps of the colimit are given by

π n+k (E k ) S →π n+k+1 (S 1 ∧E k ) π n+k+1 (s k ) +1 E k −→π n+k+1 .

Aweak equivalenceof spectra is a map between spectra, which induces an isomorphism on the stable homotopy groups.

IfAis a (pointed) topological space,Xa (pointed)C-space, thenA×X(resp.A∧X) and map(A X this notion a With) are defined objectwise.C-spectrum is the same as a family of pointedC-spaces{E n  n∈Z}together with structure mapsS 1 ∧E n →E n+1 (orE n →ΩE n+1 ). A homotopy between two (pointed)C-mapsf g:X→Yis aC-mapH:X×I→Y (resp.H:X∧I + →Y) such thatH 0 =fandH 1 =g.

Definition 1.2LetXbe a contravariant andYa covariantC-space. Define their tensor product to be the space X⊗ C Y:= a X(c)×Y(c)/∼ c∈Ob(C)

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