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

Authors: Michael A. Hill and Carolyn Yarnall
Journal: Proc. Amer. Math. Soc. 146 (2018), 3605-3614
MSC (2010): Primary 55N91, 55P91, 55Q10
Published electronically: May 4, 2018
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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 $ H$-geometric fixed points are $ (n/\vert H\vert-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 $ C_p$-spectrum and show how a very small number of functors determine all of the slices for $ C_{p^n}$-spectra.

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Additional Information

Michael A. Hill
Affiliation: Department of Mathematics, University of California Los Angeles Los Angeles, California 90025

Carolyn Yarnall
Affiliation: Department of Mathematics, California State University Dominguez Hills Carson, California 90747

Keywords: Equivariant stable homotopy theory, slice filtration, Mackey functor
Received by editor(s): April 17, 2017
Received by editor(s) in revised form: July 13, 2017
Published electronically: May 4, 2018
Additional Notes: The first author was supported by NSF Grant DMS-1509652.
Communicated by: Michael A. Mandell
Article copyright: © Copyright 2018 American Mathematical Society