Quasi-Elliptic Cohomology and its Power Operations

TL;DR

This paper explores the structure of quasi-elliptic cohomology, a variant of Tate K-theory, demonstrating how it incorporates power operations and classifies finite subgroups of the Tate curve.

Contribution

It introduces power operations in quasi-elliptic cohomology and links Tate K-theory of symmetric groups to the classification of finite subgroups of the Tate curve.

Findings

Quasi-elliptic cohomology has well-defined power operations.

Tate K-theory of symmetric groups classifies finite subgroups of the Tate curve.

The theory connects orbifold K-theory with algebraic structures of elliptic curves.

Abstract

Quasi-elliptic cohomology is a variant of Tate K-theory. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. In this paper we show how this theory is equipped with power operations. We also prove that the Tate K-theory of symmetric groups modulo a certain transfer ideal classify the finite subgroups of the Tate curve.