Comparing tempered and equivariant elliptic cohomology

TL;DR

This paper establishes a natural equivalence between two advanced equivariant cohomology theories, tempered and equivariant elliptic cohomology, using derived algebraic geometry, and demonstrates its application to fixed points of equivariant topological modular forms.

Contribution

It constructs a natural equivalence between tempered and equivariant elliptic cohomology theories and applies this to analyze fixed points of equivariant topological modular forms.

Findings

Natural equivalence between the two theories where they overlap

Demonstration of dualisability of fixed points for certain Lie groups

Reduction of the problem to an argument of Gepner--Meier

Abstract

Lurie and Gepner--Meier each define equivariant cohomology theories, namely \emph{tempered cohomology} and \emph{equivariant elliptic cohomology}, respectively, using derived algebraic geometry. We construct a natural equivalence between these theories where they overlap. Moreover, we emphasise the naturality and coherence of both these equivariant theories as well as our comparison. To demonstrate the use of this comparison, we show that the $G$-fixed points of equivariant topological modular forms is dualisable as a $\mathrm{TMF}$-module for all compact Lie groups $G$ that decompose as a product of a torus and a finite group by formally reducing to an argument of Gepner--Meier.