Algebraic K-theory of group rings and topological cyclic ... · Algebraic K-theory of group rings...

ConjecturesTheorems

Proofs

Algebraic K -theory of group ringsand topological cyclic homology

John Rognes

University of Oslo, Norway

Nordic Topology Meeting 2014

John Rognes Algebraic K -theory of group ringsand topological cyclic homology

ConjecturesTheorems

Proofs

Outline

1 Conjectures

2 Theorems

3 Proofs


ConjecturesTheorems

Proofs

Credits

This is an overview of joint work withWolfgang Lück,Holger Reich andMarco Varisco.


ConjecturesTheorems

Proofs

Outline

1 Conjectures

2 Theorems

3 Proofs


ConjecturesTheorems

Proofs

Borel conjecture

Conjecture (Borel (1953))Let G be any discrete group. Any two closed manifolds of thehomotopy type of BG are homeomorphic.

This is an analogue of the Poincaré conjecture for asphericalmanifolds.


ConjecturesTheorems

Proofs

Novikov conjecture

The integral Pontryagin classes pi(TM) ∈ H4i(M;Z) are nottopological invariants, but the rational Pontryagin classes are.Consider a map u : M → BG from an n-manifold to a classifyingspace. The higher x-signature, for x ∈ Hn−4i(BG;Q), is therational number

signx (M,u) = 〈Li(TM) ∪ u∗(x), [M]〉 .

Conjecture (Novikov (1970))

If h : M ′ → M is an orientation-preserving homotopyequivalence, then signx (M,u) = signx (M,uh).


ConjecturesTheorems

Proofs

L-theory assembly map

Conjecture (Novikov, reformulated)The L-theory assembly map

aL : BG+ ∧ L(Z) −→ L(Z[G])

is rationally injective, i.e., the induced homomorphism

aL∗ ⊗Q : H∗(BG; L∗(Z))⊗Q −→ L∗(Z[G])⊗Q

is injective in each degree.


ConjecturesTheorems

Proofs

Hsiang conjecture

Conjecture (Hsiang (1983))

If G is a torsion-free group, and BG has the homotopy type of afinite CW complex, then the K -theory assembly map

aK : BG+ ∧ K (Z) −→ K (Z[G])

is a rational equivalence.


ConjecturesTheorems

Proofs

Families

A family F of subgroups of G is a collection of subgroups,closed under conjugation with elements of G and passageto subgroups.Let EF denote the universal G-CW space with stabilizersin F . Universality amounts to the condition that EF H iscontractible for each H ∈ F .


ConjecturesTheorems

Proofs

The orbit category

DefinitionThe orbit category Or G has as objects the homogeneousG-spaces G/H, and as morphisms the G-maps.

The rule G/H 7→ EF H defines a contravariant functorEF ? from Or G to spaces.The rule G/H 7→ K (Z[H]) can be extended to a covariantfunctor K (Z[?]) from Or G to spectra.


ConjecturesTheorems

Proofs

Family assembly map

The smash product

EF+ ∧Or G K (Z[−]) = EF ?+ ∧Or G K (Z[?])

is a spectrum defined as a homotopy coend.The G-map EF → ∗ induces a natural map

aK : EF+ ∧Or G K (Z[−]) −→ ∗+ ∧Or G K (Z[−]) ' K (Z[G]) ,

which we call the K -theory assembly map for F .


ConjecturesTheorems

Proofs

Farrell–Jones conjecture

A group is called virtually cyclic if it contains a (finite or infinite)cyclic subgroup of finite index. Let G be any discrete group, andlet V Cyc be the family of virtually cyclic subgroups of G.

Conjecture (Farrell-Jones (1993))The K -theory assembly map for V Cyc,

aK : EV Cyc+ ∧Or G K (Z[−]) −→ K (Z[G]) ,

is an equivalence.


ConjecturesTheorems

Proofs

Outline

1 Conjectures

2 Theorems

3 Proofs


ConjecturesTheorems

Proofs

The Bökstedt–Hsiang–Madsen theorem

Theorem (Bökstedt–Hsiang–Madsen (1993))

Let G be a discrete group such that condition (H’) holds.(H’) H∗(BG;Z) is of finite type.Then the connective K -theory assembly map

aK : BG+ ∧ K (Z) −→ K (Z[G])

is rationally injective.


ConjecturesTheorems

Proofs

Our theorem

Theorem (Lück–Reich–Rognes–Varisco)

Let G be a discrete group such that conditions (H) and (K) holdfor each finite cyclic subgroup C of G:(H) H∗(BZGC;Z) is of finite type, where ZGC is the centralizer

of C in G;(K) The canonical map K (Z[C]) −→

∏p K (Zp[C])∧p is rationally

injective in each degree, where p ranges over all primes.Then the connective K -theory Farrell–Jones assembly map

aK : EV Cyc+ ∧Or G K (Z[−]) −→ K (Z[G])

is rationally injective.


ConjecturesTheorems

Proofs

Comments

Condition (K) is known to hold when C is the trivial group,which is why there is no explicit condition (K’) in the resultof Bökstedt–Hsiang–Madsen.Condition (K) holds in degrees t ≤ 1; in degrees t ≥ 2 it isexpected to hold in all cases, and would follow from theSchneider conjecture (1979), generalizing Leopoldt’sconjecture from K1 to Kt .Condition (H), which encompasses Condition (H’), appearsto be an intrinsic limitation of the cyclotomic trace methodas applied to this problem.


ConjecturesTheorems

Proofs

Outline

1 Conjectures

2 Theorems

3 Proofs


ConjecturesTheorems

Proofs

Reduction to finite subgroups

Let Fin be the family of finite subgroups of G.

Proposition (Grunewald (2008))The family comparison map

EFin+ ∧Or G K (Z[−]) −→ EV Cyc+ ∧Or G K (Z[−])

is a rational equivalence.


ConjecturesTheorems

Proofs

Passage to spherical group rings

Let S be the sphere spectrum.

PropositionThe linearization maps

EFin+ ∧Or G K (S[−]) −→ EFin+ ∧Or G K (Z[−])

andK (S[G]) −→ K (Z[G])

are rational equivalences.


ConjecturesTheorems

Proofs

Summary of first reductions

EV Cyc+ ∧Or G K (Z[−])aK// K (Z[G])

EFin+ ∧Or G K (Z[−])aK//

'Q

OO

K (Z[G])

EFin+ ∧Or G K (S[−])aK

//

'Q

OO

K (S[G])

'Q

OO


ConjecturesTheorems

Proofs

Topological cyclic homology

The cyclotomic trace map to topological cyclic homology givesa natural transformation

trc : K (S[−]) −→ TC(S[−]; p)

of functors from Or G to spectra.


//

1∧trc��

K (S[G])

trc��

EFin+ ∧Or G TC(S[−]; p)aTC// TC(S[G]; p)


ConjecturesTheorems

Proofs

The role of condition (K)

Proposition (Hesselholt–Madsen)

Let C be a finite group. If K (Z[C])→ K (Zp[C])∧p is rationallyinjective, then so is trc : K (S[C]) −→ TC(S[C]; p).

Proposition (Lück)If the above holds for each finite cyclic subgroup C of G, then

EFin+ ∧Or G K (S[−]) −→ EFin+ ∧Or G TC(S[−]; p)

is also rationally injective.


ConjecturesTheorems

Proofs

A warning

In the case of the trivial family F = {e}, the lower horizontalmap

aTC : BG+ ∧ TC(S; p) −→ TC(S[G]; p)

is the TC-assembly map considered by [BHM].It does not split quite as claimed in Madsen’s survey (1994).


ConjecturesTheorems

Proofs

The homotopy pullback square

There is a homotopy Cartesian square

TC(S[G]; p)α //

β��

C(S[G]; p)

trf��

THH(S[G])1−∆p

// // THH(S[G])

where the Bökstedt–Hsiang–Madsen functor

C(S[G]; p) = holimn≥1

THH(S[G])hCpn

is the homotopy limit over the transfer maps.


ConjecturesTheorems

Proofs

Explanations

The composite β ◦ trc : K (S[G])→ THH(S[G]) is theWaldhausen trace map, in the form given by Bökstedt.There is a natural equivalence

THH(S[G]) ' S[Bcy (G)] ,

where Bcy (G) is the cyclic bar construction on G.∆p : Bcy (G)→ Bcy (G) is the p-th power map.


ConjecturesTheorems

Proofs

A decomposition

There is a decomposition

Bcy (G) =∐[g]

Bcy[g](G)

where [g] ranges over the conjugacy classes of elementsin G, and Bcy

[g](G) is the path component that contains thevertex g.The p-th power map ∆p takes Bcy

[g](G) to Bcy[gp](G).


ConjecturesTheorems

Proofs

The difficulty

The THH-assembly map

aTHH : BG+ ∧ THH(S) −→ THH(S[G])

is induced by the inclusion BG ∼= Bcye (G)→ Bcy (G).

It is split by the evident retraction pr : Bcy (G)+ → BG+, but

pr : THH(S[G]) −→ BG+ ∧ THH(S)

is not in general compatible with the p-th power map ∆p.This does not produce a map

pr : TC(S[G]; p) −→ BG+ ∧ TC(S; p)

splitting the TC-assembly map.


ConjecturesTheorems

Proofs

The solution

The original, correct, strategy of Bökstedt–Hsiang–Madsen,does not split the assembly map aTC but the assembly map

aC : BG+ ∧ C(S; p) −→ C(S[G]; p)

for the functor C.Hence we must construct a natural transformation

α : TC(S[−]; p) −→ C(S[−]; p)

of functors from Or G to spectra.This requires natural Segal–tom Dieck splittings and Adamstransfer equivalences, constructed by Reich–Varisco (2014).


ConjecturesTheorems

Proofs

Reduction to C and THH

EFin+ ∧Or G TC(S[−]; p)aTC

//

1∧(α∨β)��

TC(S[G]; p)

α∨β��

EFin+ ∧Or G (C(S[−]; p) ∨ T (S[−]))aC∨T

// C(S[G]; p) ∨ T (S[G])

(We sometimes abbreviate THH to T on this page, and thenext.)


ConjecturesTheorems

Proofs

Rational injectivity of α ∨ β

Proposition ([BHM])Let D be a finite group. The map

α ∨ β : TC(S[D]; p) −→ C(S[D]; p) ∨ THH(S[D])

is rationally injective in non-negative degrees.

Proposition (Lück)The map

EFin+∧Or G TC(S[−]; p) −→ EFin+∧Or G (C(S[−]; p)∨T (S[−]))

is rationally injective in non-negative degrees.


ConjecturesTheorems

Proofs

The F -part of THH

For any family F let

BcyF (G) =

∐〈g〉∈F

Bcy[g](G)

be the union of the path components in Bcy (G) that contain thevertices (g) such that the cyclic group 〈g〉 is a member of thefamily F .The F -part of THH(S[G]) satisfies

THHF (S[G]) ' S[BcyF (G)] .


ConjecturesTheorems

Proofs

Splitting the THH-assembly map for F

The inclusion BcyF (G)→ Bcy (G), and the projection

prF : Bcy (G)+ → BcyF (G)+ make the F -part a retract of

THH(S[−]).

PropositionThe left hand vertical map and the lower horizontal map in thecommutative square

EF+ ∧Or G THH(S[−])aTHH

//

1∧prF '��

THH(S[G])

prF��

EF+ ∧Or G THHF (S[−])aTHHF

'// THHF (S[G])

are stable equivalences.


ConjecturesTheorems

Proofs

Splitting the C-assembly map for F

EF+ ∧Or G C(S[−]; p)

aC

++

κ

��

holimn(EF+ ∧Or G THH(S[−])hCpn

)//

'��

C(S[G]; p)

prF��

holimn(EF+ ∧Or G THHF (S[−])hCpn

)'

// CF (S[G]; p)


ConjecturesTheorems

Proofs

End of proof

Proposition (Lück–Reich–Varisco (2003))Assuming condition (H),

κ : EF+ ∧Or G C(S[−]; p) −→ holimn

(EF+ ∧Or G THH(S[−])hCpn

)is an equivalence for F the family of finite cyclic subgroupsof G, hence also for the family Fin of finite subgroups of G.

This implies that aC for Fin is split injective. Q.E.D.


ConjecturesTheorems

Proofs

Summary of all reductions (T = THH)

EV Cyc+ ∧Or G K (Z[−])aK

// K (Z[G])

EFin+ ∧Or G K (Z[−])aK

//

'Q

OO

K (Z[G])


//

'Q

OO

Q-inj.��

K (S[G])

'Q

OO

trc��

EFin+ ∧Or G TC(S[−]; p)aTC

//

non-neg. Q-inj.��

TC(S[G]; p)

α∨β��

EFin+ ∧Or G (C(S[−]; p) ∨ T (S[−]))aC∨T

inj.// C(S[G]; p) ∨ T (S[G])


Algebraic K-theory of group rings and topological cyclic ... · Algebraic K-theory of group rings...

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