Equivariant homotopy equivalence of group representations

Kawakubo, Katsuo

doi:10.2969/jmsj/03210105

Cited by 12 publications

(5 citation statements)

References 7 publications

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“…Let λ i be the T-representation where z ∈ T ⊂ C × acts as z i . These exhaust the nontrivial irreducible real representations of T. The representation spheres S λ i and S λ j are all integrally inequivalent [Kaw80], but in the p-local setting we have S λ i ∼ = S λ j whenever the p-adic valuations of i and j agree. Thus, let λ r = λ p r .…”

Section: Mackey Functorsmentioning

confidence: 99%

A slice refinement of Bökstedt periodicity

Sulyma

2020

Preprint

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Let R be a perfectoid ring. Hesselholt and Bhatt-Morrow-Scholze have identified the Postnikov filtration on THH(R; Zp): it is concentrated in even degrees, generated by powers of the Bökstedt generator σ, generalizing classical Bökstedt periodicity for R = Fp. We study an equivariant generalization of the Postnikov filtration, the regular slice filtration, on THH(R; Zp). The slice filtration is again concentrated in even degrees, generated by RO(T)-graded classes which can loosely be thought of as the norms of σ. The slices are expressible as RO(T)-graded suspensions of Mackey functors obtained from the Witt Mackey functor. We obtain a sort of filtration by q-factorials. A key ingredient, which may be of independent interest, is a close connection between the Hill-Yarnall characterization of the slice filtration and Anschütz-le Bras' q-deformation of Legendre's formula.

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Section: Mackey Functorsmentioning

confidence: 99%

A slice refinement of Bökstedt periodicity

Sulyma

2020

Preprint

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“…The representations λ(k) for 1 ≤ k ≤ (p−1)/2 are representatives for the isomorphism classes of non-trivial real representations of C p n . Work of Kawakubo shows that the representation spheres are all integrally inequivalent [6]. Even so, the exact choices are immaterial due to the following observation.…”

Section: Slices For Cyclic P-groupsmentioning

confidence: 99%

A new formulation of the equivariant slice filtration with applications to $C_p$-slices

Hill

Yarnall

2018

Proc. Amer. Math. Soc.

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Abstract. This paper provides a new way to understand the equivariant slice filtration. We give a new, readily checked condition for determining when a G-spectrum is slice n-connective. In particular, we show that a G-spectrum is slice greater than or equal to n if and only if for all subgroups H, the Hgeometric fixed points are (n/|H| − 1)-connected. We use this to determine when smashing with a virtual representation sphere S V induces an equivalence between various slice categories. Using this, we give an explicit formula for the slices for an arbitrary Cp-spectrum and show how a very small number of functors determine all of the slices for C p n -spectra.

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“…The first step is to identify all λ(rp k ) for p r. If we localize at p, then all S λ(rp k ) are homotopy equivalent to each other, since the degree r map is invertible now. If we don't localize, then by [Kaw80], different S λ(rp k ) are not even stably equivalent. However, we have the following.…”

Section: Equivariant Orthogonal Spectramentioning

confidence: 99%

Equivariant Eilenberg-Mac Lane spectra in cyclic $p$-groups

Zeng

2017

Preprint

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In this paper we compute RO(G)-graded homotopy Mackey functors of HZ, the Eilenberg-Mac Lane spectrum of the constant Mackey functor of integers for cyclic p-groups and give a complete computation for C p 2 . We also discuss homological algebra of Z-modules for cyclic p-groups, and interactions between these two. The goal of computation in this paper is to understand various slice spectral sequences as RO(G)-graded spectral sequences of Mackey functors. Contents 1. Introduction 1 2. Mackey Functors and Z-Modules 4 3. Homological Algebra of Z-Modules 12 4. Equivariant Orthogonal Spectra 20 5. HZ and its modules 28 6. Computation of π (HZ) for C p 2 32 References 50

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Equivariant homotopy equivalence of group representations

Cited by 12 publications

References 7 publications

A slice refinement of Bökstedt periodicity

A slice refinement of Bökstedt periodicity

A new formulation of the equivariant slice filtration with applications to $C_p$-slices

Equivariant Eilenberg-Mac Lane spectra in cyclic $p$-groups

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