Rigidity and exotic models for $v_1$-local $G$-equivariant stable homotopy theory

Irakli Patchkoria; Constanze Roitzheim

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arXiv · 1705.03855

Rigidity and exotic models for $v_1$-local $G$-equivariant stable homotopy theory

Irakli Patchkoria·Constanze Roitzheim

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Abstract

We prove that the $v_1$-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 $v_1$-local $G$-equivariant stable homotopy category, showing that for primes $p \ge 5$ equivariant rigidity fails in general.

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BibTeX RIS

Irakli Patchkoria, Constanze Roitzheim. 2017-05-10. Rigidity and exotic models for $v_1$-local $G$-equivariant stable homotopy theory. https://arxiv.org/abs/1705.03855

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