Operations on Nil-terms

Joachim Grunewald

Rheinische Friedrich Wilhelms-University Bonnhttp://www.math.uni-bonn.de/people/grunewal/

NIL PHENOMENA IN TOPOLOGYAPRIL 15, 2007

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Main Result (G., John Klein, Tibor Macko 2007)

If p is an odd prime the following holds.

1

π2p−2 NA(∗)∧p ∼= ⊕n∈N×Fpβ±nπ2p−1 NA(∗)∧p ∼= ⊕n∈N×Fpγ±n

πi NA(∗)∧p ∼= 0 for i < 2p − 2 , 2p − 1 < i ≤ 4p − 7.

2 The Zp[N×]-module structure on π∗ NA(∗)∧p is given by

(n, βm) 7→ βnm

(n, γm) 7→ n · γnm

Corollary

The Zp[N×]-module π2p−2 NA(∗)∧p is (finitely) generated by β±1, and theZp[N×]-module π2p−1 NA(∗)∧p is not finitely generated.

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Outline

Frobenius and Verschiebung operations on NK

Frobenius and Verschiebung operations on NA

The proof of the main result

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Operations on NK

Theorem (Bass, Heller, Swan 1964; Quillen 1976)

Let R be a ring. For every i ∈ Z we have

Ki (R[t, t−1]) = Ki (R)⊕ Ki−1(R)⊕ NKi (R)⊕ NKi (R).

whereNKi (R) = Ker

(ε : Ki (R[t]) → Ki (R)

).

Definition (Nili)

Objects of the category NIL(R) are pairs (P, ν), where P is a finitelygenerated projective module and ν is a nilpotent endomorphism of P.A morphism f : (P, ν) → (P ′, ν ′) in NIL(R) is a morphism f : P → P ′

such that ν ′ f = f ν.

Let π : Kn

(NIL(R)

)→ Kn(R) be the projection. We define

Nili (R) := Ker(π : Kn(NIL(R)) → Kn(R)

).

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Definition

For a natural number n we define

ϕn : R[t] → R[t]

t 7→ tn.

We define the Verschiebung to be the induced map

Vn : NKi (R) → NKi (R)

and the Frobenius to be the corresponding transfer map:

Fn : NKi (R) → NKi (R).

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Definition

For a natural number n we define the Verschiebung to be the map which isinduced by the functor

Vn : NIL(R) → NIL(R)

(P, ν) 7→ (Pn,

0 ν

id. . .. . . 0

id 0

).

We define the Frobenius to be the map which is induced by the functor

Fn : NIL(R) → NIL(R)

(P, ν) 7→ (P, νn).

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Theorem (Stienstra 1982)

The two definitions of the Frobenius and Verschiebung operation coincide.

Theorem (Stienstra 1982)

The Frobenius and Verschiebung operations satisfy the following identities:

1 V1 = F1 = id

2 FnVn = n

3 VnVm = Vn·m4 FnFm = Fn·m5 VnFm = FmVn if (n,m) = 1

6 For every x ∈ NKi (R) there exist an N ∈ N such that Fn(x) = 0 forall n ≥ N.

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Conclusion out of the operations

Theorem (Weibel 1980)

Let G be a finite group of order n. The groups NKi (ZG ) are n-primarytorsion.

Theorem (Farrell 1978)

Let R be a ring the groups NKi (R) are either trivial or not finitelygenerated as an abelian group.

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Since Vn Vm = Vn·m we obtain an ZN×-module structure on NKi (R).

Theorem (Connolly, da Silva 1997)

Let G be a finite group. The group NK0(ZG ) is a finitely generatedZN×-module.

Theorem (Loday, Guin-Walery 1981)

The group NK2(ZCp) is a cyclic ZN×-module.

Question

Let G be a finite group. Are the groups NKi (ZG ) always finitely generatedZN×-modules?

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Operations on NA

Theorem (Huttemann, Klein, Vogell, Waldhausen, Williams 2001)

There is a functorial decomposition

Afd(X × S1) = Afd(X )× BAfd(X )× NA(X )× NA(X ).

Here, Afd(X ) is a version of A(X ) that is based on finitely dominatedspaces and BAfd(X ) is a certain canonical non-connective delooping ofAfd(X ).

Definition (Nili)

Objects of NIL(∗,M) are pairs (Y , f ), where Y is in Cfd(M) and f isan M-map f : Y → Y with an n ∈ N such that f n is equivariantly nullhomotopic. A morphism f : (Y , f ) → (Y ′, f ′) in NIL(∗,M) is amorphism g : Y → Y ′ such that f ′ g = g f .

Let π : Kn

(NIL(∗,M)

)→ Kn(Cfd(M)) be the projection. We define

Nili (∗,M) := Ker(π : Ki (NIL(∗,M)) → Ki (Cfd(M))

).

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Definition

For a natural number n we define

ϕn : S1 → S1

to be the n-fold cover map. We define the Verschiebung to be the inducedmap

Vn : NA(X ) → NA(X )

and the Frobenius to be the corresponding transfer map:

Fn : NA(X ) → NA(X ).

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Definition

For a natural number n we define the Verschiebung to be the map which isinduced by the functor

Vn : NILfd(∗,M) → NILfd(∗,M)

(Y , f ) 7→ (∨nY ,

0 f

id. . .. . . 0

id 0

).

We define the Frobenius to be the map induced by the functors

Fn : NILfd(∗,M) → NILfd(∗,M)

(Y , f ) 7→ (Y , f n).

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Theorem (G., Klein, Macko)

The two definitions of the Frobenius and Verschiebung operation coincide.

Theorem (G., Klein, Macko)

The Frobenius and Verschiebung operations satisfy the following identities:

1 V1 = F1 = id

2 FnVn = n

3 VnVm = Vn·m4 FnFm = Fn·m5 VnFm = FmVn if (n,m) = 1

6 There exist an N ∈ N for every x ∈ πi NA(X ) such that Fn(x) = 0 forall n ≥ N.

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Conclusion out of the operations

Corollary

The groups πi NA(X ) and all of its p-primary subgroups are either trivialor not finitely generated as an abelian group.

Question

Is there a Witt vector module structure on πi NA(X )?

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Proof of the main result

Main Result (G., John Klein, Tibor Macko)

If p is an odd prime the following holds.

1

π2p−2 NA(∗)∧p ∼= ⊕n∈N×Fpβ±nπ2p−1 NA(∗)∧p ∼= ⊕n∈N×Fpγ±n

πi NA(∗)∧p ∼= 0 for i < 2p − 2 , 2p − 1 < i ≤ 4p − 7.

2 The Zp[N×]-module structure on π∗ NA(∗)∧p is given by

(n, βm) 7→ βnm

(n, γm) 7→ n · γnm

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General strategy to study NA(X )

S1+ ∧ F (∗) //

F (S1) //

NA(∗)× NA(∗)

S1+ ∧ A(∗) //

A(S1) //

NA(∗)× NA(∗)

S1+ ∧ K (Z) // K (Z[t, t−1]) // NK(Z)× NK(Z)

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Proposition

Let p be an odd prime. Then there is a (4p − 7)-connected map∨n∈Z

Σ2p−2HFp ∧ (S1+) −→ F (S1)∧p .

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For a space X we denote Q(X ) := Ω∞Σ∞X and we let SG = Q(S0)1 bethe identity component which is a topological monoid with respect to thecomposition product.

Proposition (1)

If p is an odd prime, then there is a (4p − 7)-cartesian square

A(BSG × S1)∧p //

A(S1)∧p

A(S1)∧p // K (Z[t, t−1])∧p ,

where the left vertical arrow is induced by the projection map and theright vertical arrow is the linearization map.

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We denote ΛX = Map(S1,X ) and ev: ΛX → X is the evaluation map.For a map f : Y → X the symbol Λ(f : Y → X ) denotes the pullback ofthe diagram

Yf // X ΛX .

evoo

Goodwillie constructs a natural map A(X ) → Σ∞+ ΛX . Given a map of

spaces f : Y → X the composition A(Y ) → A(X ) → Σ∞+ ΛX factors

through Σ∞+ Λ(f : Y → X ). This factorization has the following property:

Theorem (Goodwillie)

If f : Y → X is a k-connected map, then the square

A(Y ) //

Σ∞+ Λ(f : Y → X )

A(X ) // Σ∞+ ΛX

is (2k − 1)-cartesian.

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In the given situation we obtain:

hofiber(A(BSG × S1) → A(BSG ))∧p → Σ∞BSG∧p ∧ (ΛS1

+).

We have a decomposition

ΛS1 '∐n∈Z

S1(n),

further we have a (4p − 6)-connected map

BSG∧p → Σ2p−2HFp.

Thus we obtain a 4p − 7-conected map∨n∈Z

Σ2p−2HFp ∧ (S1+) −→ F (S1)∧p .

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Proof of Proposition 1

Lemma

There is a (4p − 6)-cartesian square of S-algebras

S[SG × Z]∧pθ //

ε

S[Z]∧p

l

S[Z]∧p l// HZ[Z]∧p .

Recall

1 K (S[SG × Z]) = A(BSG × S1),

2 K (S[Z]) = A(S1),

3 K (HZ[Z]) = K (Z[t, t−1]).

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