The K\"unneth Theorem in equivariant K-theory for actions of a cyclic group of order 2

TL;DR

This paper addresses the failure of the K"unneth Theorem in equivariant K-theory for cyclic groups of order 2 and proposes a solution using RO(G)-graded equivariant K-theory.

Contribution

It introduces a method to restore the K"unneth Theorem in the case of G = Z/2 by leveraging RO(G)-graded equivariant K-theory.

Findings

Restores the K"unneth Theorem for G = Z/2 using RO(G)-grading.

Demonstrates the failure of the classical theorem in this setting.

Provides a new framework for equivariant K-theory with finite groups.

Abstract

The K\"unneth Theorem for equivariant (complex) K-theory K^*_G, in the form developed by Hodgkin and others, fails dramatically when G is a finite group, and even when G is cyclic of order 2. We remedy this situation in this very simplest case G = Z/2 by using the power of RO(G)-graded equivariant K-theory.