$C_2$-equivariant topological modular forms

TL;DR

This paper computes the homotopy groups of the $C_2$ fixed points of equivariant topological modular forms at the prime 2, revealing their structure as a tensor product with a finite cell complex.

Contribution

It provides the first explicit computation of these homotopy groups and describes their module structure over TMF, using the descent spectral sequence.

Findings

Homotopy groups of $C_2$ fixed points computed at prime 2.

Identifies the module structure as a tensor product with a finite cell complex.

Uses descent spectral sequence for the computation.

Abstract

We compute the homotopy groups of the $C_2$ fixed points of equivariant topological modular forms at the prime $2$ using the descent spectral sequence. We then show that as a $\mathrm{TMF}$-module, it is isomorphic to the tensor product of $\mathrm{TMF}$ with an explicit finite cell complex.