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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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