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

## TL;DR

This paper proves the uniqueness of the $v_1$-local $G$-equivariant stable homotopy category at prime 2 and constructs exotic models at primes 5 and above, revealing when rigidity fails.

## Contribution

It establishes the $G$-equivariant rigidity at prime 2 and constructs algebraic and $G$-equivariant exotic models for primes at least 5, showing the limits of rigidity.

## Key findings

- Unique $G$-equivariant model at $p=2$
- Existence of algebraic exotic models at $p \\ge 5$
- Rigidity fails for $p \\ge 5$ in general

## 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.