Classifying spaces for families (Lecture III)

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Classifying spaces for families (Lecture III)

Wolfgang Lück Münster Germany

email lueck@math.uni-muenster.de http://www.math.uni-muenster.de/u/lueck/

Hangzhou, July 2007

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Flashback

We have introduced theFarrell-Jones Conjectureand the Baum-Connes Conjecturefor torsionfree groups:

H_n(BG;K_R) −^∼⁼→ K_n(RG);

Hn(BG;L^h−∞i_R ) −^∼⁼→ L^h−∞i_n (RG);

Kn(BG) −^∼⁼→ Kn(C_r^∗(G)).

We have discussed applications of these conjectures such as to theKaplansky Conjectureand theBorel Conjecture.

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Flashback

H_n(BG;K_R) −^∼⁼→ K_n(RG);

Hn(BG;L^h−∞i_R ) −^∼⁼→ L^h−∞i_n (RG);

Kn(BG) −^∼⁼→ Kn(C_r^∗(G)).

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Flashback

H_n(BG;K_R) −^∼⁼→ K_n(RG);

Hn(BG;L^h−∞i_R ) −^∼⁼→ L^h−∞i_n (RG);

Kn(BG) −^∼⁼→ Kn(C_r^∗(G)).

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Flashback

H_n(BG;K_R) −^∼⁼→ K_n(RG);

Hn(BG;L^h−∞i_R ) −^∼⁼→ L^h−∞i_n (RG);

Kn(BG) −^∼⁼→ Kn(C_r^∗(G)).

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Cliffhanger

Question (Classifying spaces for families)

Is there a versionEF(G)of the classifying space EG which takes the structure of the family of finite subgroups or other familiesF of subgroups into account and can be used for a general formulation of the Farrell-Jones Conjecture?

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Cliffhanger

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Cliffhanger

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Outline

We introduce the notion of theclassifying space of a familyF of subgroupsEF(G)andJF(G).

In the case, whereF is the familyCOMof compact subgroups, we present some nice geometric models forEF(G)and explain E_F(G)'J_F(G).

We discussfiniteness propertiesof these classifying spaces.

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Outline

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Outline

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Outline

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Classifying spaces for families of subgroups

Definition (G-CW-complex)

AG-CW -complexX is aG-space together with aG-invariant filtration

∅=X₋₁⊆X₀⊆. . .⊆X_n⊆. . .⊆ [

n≥0

X_n=X

such thatX carries thecolimit topologywith respect to this filtration, andX_nis obtained fromX_n−1for eachn≥0 byattaching equivariant n-dimensional cells, i.e., there exists aG-pushout

`

i∈InG/H_i×Sⁿ⁻¹

‘

i∈Inq_iⁿ

//

X_n−1

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Classifying spaces for families of subgroups

∅=X₋₁⊆X₀⊆. . .⊆X_n⊆. . .⊆ [

n≥0

X_n=X

`

‘

i∈Inq_iⁿ

//

X_n−1

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Classifying spaces for families of subgroups

∅=X₋₁⊆X₀⊆. . .⊆X_n⊆. . .⊆ [

n≥0

X_n=X

`

‘

i∈Inq_iⁿ

//

X_n−1

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Classifying spaces for families of subgroups

∅=X₋₁⊆X₀⊆. . .⊆X_n⊆. . .⊆ [

n≥0

X_n=X

`

‘

i∈Inq_iⁿ

//

X_n−1

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Classifying spaces for families of subgroups

∅=X₋₁⊆X₀⊆. . .⊆X_n⊆. . .⊆ [

n≥0

X_n=X

`

‘

i∈Inq_iⁿ

//

X_n−1

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Group meanslocally compact Hausdorff topological group with a countable basis for its topology, unless explicitly stated differently.

Example (Simplicial actions)

LetX be a simplicial complex. Suppose thatGacts simplicially onX. ThenGacts simplicially also on thebarycentric subdivisionX⁰, and all isotropy groups are open and closed. TheG-spaceX⁰ inherits the structure of aG-CW-complex.

Example (Smooth actions)

LetGbe a Lie group acting properly and smoothly on a smooth manifoldM.

ThenM inherits the structure ofG-CW-complex.

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Definition (ProperG-action)

AG-spaceX is calledproperif for each pair of pointsx andy inX there are open neighborhoodsVx ofx andWy ofy inX such that the closure of the subset{g ∈G|gV_x∩W_y 6=∅}ofGis compact.

Lemma

A proper G-space has always compact isotropy groups.

A G-CW -complex X is proper if and only if all its isotropy groups are compact.

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Lemma

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Lemma

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Lemma

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Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of (closed) subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

VCYC = {virtually cyclic subgroups};

COM = {compact subgroups};

COMOP = {compact open subgroups};

ALL = {all subgroups}.

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Examples forF are:

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Examples forF are:

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Examples forF are:

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Examples forF are:

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Examples forF are:

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Examples forF are:

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Examples forF are:

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Examples forF are:

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Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG. A model for theclassifying G-CW -complex for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofEF(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →E_F(G).

We abbreviateE G:=ECOM(G)and call it theuniversal G-CW -complex for proper G-actions.

We also writeEG=ET R(G).

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Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG.A model for theclassifying G-CW -complex for the familyF is aG-CW-complexEF(G)which has the following properties:

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Theorem (Homotopy characterization ofEF(G)) LetF be a family of subgroups.

There exists a model for EF(G)for any familyF;

Two model for EF(G)are G-homotopy equivalent;

A G-CW -complex X is a model for E_F(G)if and only if all its isotropy groups belong toF and for each H∈ F the H-fixed point set X^H is weakly contractible.

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A model forE_ALL(G)isG/G;

EG→BG:=G\EGis theuniversalG-principal bundlefor G-CW-complexes.

Example (Infinite dihedral group)

LetD_∞=Z o Z/2=Z/2∗Z/2 be the infinite dihedral group.

A model forED∞is the universal covering ofRP^∞∨RP^∞. A model forE D∞isRwith the obviousD∞-action.

Lemma

If G is totally disconnected, then ECOMOP(G) =E G.

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Lemma

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Lemma

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Lemma

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Lemma

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Lemma

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Lemma

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Definition (F-numerableG-space)

AF-numerable G-spaceis aG-space, for which there exists an open covering{U_i |i ∈I}byG-subspaces satisfying:

For eachi∈Ithere exists aG-mapU_i →G/G_i for someG_i ∈ F; There is a locally finite partition of unity{e_i |i∈I}subordinate to {U_i |i ∈I}byG-invariant functionse_i:X →[0,1].

Notice that we do not demand that the isotropy groups of a F-numerableG-space belong toF.

Iff:X →Y is aG-map andY isF-numerable, thenX is also F-numerable.

AG-CW-complex isF-numerable if and only if each isotropy group appears as a subgroup of an element inF.

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There is also a versionJ_F(G)of a classifying space for F-numerableG-spaces.

It is characterized by the property thatJF(G)isF-numerable and for everyF-numerableG-spaceY there is up toG-homotopy precisely oneG-mapY →JF(G).

We abbreviateJG=JCOM(G)andJG=JT R(G).

JG→G\JGis theuniversalG-principal bundlefor numerable free properG-spaces.

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Theorem (Comparison ofEF(G)andJF(G),L. (2005)) There is up to G-homotopy precisely one G-map

φ:E_F(G)→J_F(G);

It is a G-homotopy equivalence if one of the following conditions is satisfied:

Each element inFis open and closed;

G is discrete;

FisCOM;

Let G be totally disconnected. Then EG→JG is a G-homotopy equivalence if and only if G is discrete.

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φ:E_F(G)→J_F(G);

G is discrete;

FisCOM;

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φ:E_F(G)→J_F(G);

G is discrete;

FisCOM;

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φ:E_F(G)→J_F(G);

G is discrete;

FisCOM;

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φ:E_F(G)→J_F(G);

G is discrete;

FisCOM;

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φ:E_F(G)→J_F(G);

G is discrete;

FisCOM;

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φ:E_F(G)→J_F(G);

G is discrete;

FisCOM;

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Special models for E G

We want to illustrate that the spaceE G=JGoften hasvery nice geometric modelsandappear naturally in many interesting situations.

LetC₀(G)be the Banach space of complex valued functions ofG vanishing at infinity with the supremum-norm. The groupGacts isometrically onC₀(G)by(g·f)(x) :=f(g⁻¹x)forf ∈C₀(G)and g,x ∈G.

LetPC₀(G)be the subspace ofC₀(G)consisting of functionsf such thatf is not identically zero and has non-negative real numbers as values.

Theorem (Operator theoretic model,Abels (1978))

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Special models for E G

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Special models for E G

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Special models for E G

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Special models for E G

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Special models for E G

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Theorem

Let G be discrete. A model for JG is the space X_G =

f:G→[0,1]

f has finite support, X

g∈G

f(g) =1

with the topology coming from the supremum norm.

Theorem (Simplicial Model)

Let G be discrete. Let P∞(G)be the geometric realization of the simplicial set whose k -simplices consist of(k+1)-tupels

(g₀,g₁, . . . ,g_k)of elements g_iin G. This is a model for E G.

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Theorem

f:G→[0,1]

g∈G

f(g) =1

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Theorem

f:G→[0,1]

g∈G

f(g) =1

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Theorem

f:G→[0,1]

g∈G

f(g) =1

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Theorem

f:G→[0,1]

g∈G

f(g) =1

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Theorem

f:G→[0,1]

g∈G

f(g) =1

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The spacesX_G andP_∞(G)have the same underlying sets but in general they have different topologies.

The identity map induces aG-mapP∞(G)→X_Gwhich is a

G-homotopy equivalence, but in general not aG-homeomorphism.

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The spacesX_G andP_∞(G)have the same underlying sets but in general they have different topologies.

The identity map induces aG-mapP∞(G)→X_Gwhich is a

G-homotopy equivalence, but in general not aG-homeomorphism.

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Theorem (Almost connected groups,Abels (1978).)

Suppose that G isalmost connected, i.e., the group G/G⁰is compact for G⁰the component of the identity element.

Then G contains a maximal compact subgroup K which is unique up to conjugation, and the G-space G/K is a model for JG.

As a special case we get:

Theorem (Discrete subgroups of almost connected Lie groups) Let L be a Lie group with finitely many path components.

Then L contains a maximal compact subgroup K which is unique up to conjugation, and the L-space L/K is a model for E L.

If G ⊆L is a discrete subgroup of L, then L/K with the obvious left

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Theorem (Actions on CAT(0)-spaces)

Let G be a (locally compact Hausdorff) topological group. Let X be a proper G-CW -complex. Suppose that X has the structure of a complete simply connectedCAT(0)-space for which G acts by isometries.

Then X is a model for E G.

The result above contains as special caseisometricGactions on simply-connected complete Riemannian manifolds with

non-positive sectional curvatureandG-actions on trees.

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Theorem (Affine buildings)

Let G be a totally disconnected group. Suppose that G acts on the affine buildingΣby simplicial automorphisms such that each isotropy group is compact.

ThenΣis a model for both JCOMOP(G)and JG and the barycentric subdivisionΣ⁰is a model for both ECOMOP(G)and E G.

An important example is the case of areductivep-adic algebraic groupGand its associatedaffine Bruhat-Tits buildingβ(G).

Thenβ(G)is a model forJGandβ(G)⁰is a model forE Gby the previous result.

For more information about buildings we refer to the lectures of Abramenko.

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TheRips complexP_d(G,S)of a groupGwith a symmetric finite setS of generators for a natural numberd is the geometric realization of the simplicial set whose set ofk-simplices consists of(k+1)-tuples(g₀,g₁, . . .g_k)of pairwise distinct elements g_i ∈Gsatisfyingd_S(g_i,g_j)≤d for alli,j∈ {0,1, . . . ,k}.

The obviousG-action by simplicial automorphisms onP_d(G,S) induces aG-action by simplicial automorphisms on the

barycentric subdivisionP_d(G,S)⁰.

Theorem (Rips complex,Meintrup-Schick (2002))

Let G be a discrete group with a finite symmetric set of generators.

Suppose that(G,S)isδ-hyperbolic for the real numberδ ≥0. Let d be a natural number with d ≥16δ+8.

Then the barycentric subdivision of the Rips complex P_d(G,S)⁰ is a

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TheRips complexP_d(G,S)of a groupGwith a symmetric finite setS of generators for a natural numberd is the geometric realization of the simplicial set whose set ofk-simplices consists of(k+1)-tuples(g₀,g₁, . . .g_k)of pairwise distinct elements g_i ∈Gsatisfyingd_S(g_i,g_j)≤d for alli,j∈ {0,1, . . . ,k}.

The obviousG-action by simplicial automorphisms onP_d(G,S) induces aG-action by simplicial automorphisms on the

barycentric subdivisionP_d(G,S)⁰.

Theorem (Rips complex,Meintrup-Schick (2002))

Let G be a discrete group with a finite symmetric set of generators.

Suppose that(G,S)isδ-hyperbolic for the real numberδ ≥0. Let d be a natural number with d ≥16δ+8.

Then the barycentric subdivision of the Rips complex P_d(G,S)⁰ is a