Rigidity and exotic models for v1-local G-equivariant stable homotopy

Irakli Patchkoria, Constanze Roitzheim (Corresponding Author)

Research output: Contribution to journalArticle

Abstract

We prove that the v1-local G-equivariant stable homotopy category for G
a finite group has a unique G-equivariant model at p = 2. This means that at the
prime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p ≥ 5 equivariant rigidity fails in general.
Original languageEnglish
JournalMathematische Zeitschrift
Early online date12 Aug 2019
DOIs
Publication statusE-pub ahead of print - 12 Aug 2019

Fingerprint

Stable Homotopy
Equivariant
Rigidity
Model
Local Spectrum
Homotopy Theory
K-theory
Homotopy
Divides
Finite Group
Fixed point
Equivalence

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Rigidity and exotic models for v1-local G-equivariant stable homotopy. / Patchkoria, Irakli; Roitzheim, Constanze (Corresponding Author).

In: Mathematische Zeitschrift, 12.08.2019.

Research output: Contribution to journalArticle

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abstract = "We prove that the v1-local G-equivariant stable homotopy category for Ga finite group has a unique G-equivariant model at p = 2. This means that at theprime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p ≥ 5 equivariant rigidity fails in general.",
author = "Irakli Patchkoria and Constanze Roitzheim",
note = "The first author acknowledges the support of the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92), the German Research Foundation Schwerpunktprogramm 1786 and the Shota Rustaveli National Science Foundation Grant 217–614. He also thanks the Hausdorff Research Institute for Mathematics in Bonn for their hospitality. Finally the first author would like to thank Gijs Heuts, Akhil Mathew, Stefan Schwede and Christian Wimmer for helpful conversations. The second author thanks the Department of Mathematical Sciences in Copenhagen and the Hausdorff Research Institute for Mathematics in Bonn for their hospitality. Furthermore, the second author would like to thank David Barnes, Anna Marie Bohmann, John Greenlees and Mike Hill for useful discussions. We thank the referee for useful comments.",
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