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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
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
ConjecturesTheorems
Proofs
Credits
This is an overview of joint work withWolfgang Lück,Holger Reich andMarco Varisco.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
ConjecturesTheorems
Proofs
Outline
1 Conjectures
2 Theorems
3 Proofs
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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).
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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 .
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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 .
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
ConjecturesTheorems
Proofs
Outline
1 Conjectures
2 Theorems
3 Proofs
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
ConjecturesTheorems
Proofs
Outline
1 Conjectures
2 Theorems
3 Proofs
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
EFin+ ∧Or G K (S[−])aK
//
1∧trc��
K (S[G])
trc��
EFin+ ∧Or G TC(S[−]; p)aTC// TC(S[G]; p)
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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).
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
ConjecturesTheorems
Proofs
The homotopy pullback square
There is a homotopy Cartesian square
TC(S[G]; p)α //
�
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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).
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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).
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.)
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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)] .
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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)
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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.
John Rognes Algebraic K -theory of group ringsand topological cyclic homology
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])
EFin+ ∧Or G K (S[−])aK
//
'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])
John Rognes Algebraic K -theory of group ringsand topological cyclic homology