# Freed-Moore K-theory

## TL;DR

This paper develops a unified framework for Freed-Moore K-theory, extending twisted equivariant K-theory to include Real K-theory and variants, establishing foundational properties and formulations.

## Contribution

It formulates Freed-Moore K-theory using Fredholm operators and Karoubi's gradations, clarifying their relationships and establishing key properties like Bott periodicity.

## Key findings

- Established Bott periodicity for Freed-Moore K-theory
- Proved Thom isomorphism in this framework
- Unified various formulations of the theory

## Abstract

The twisted equivariant K-theory given by Freed and Moore is a K-theory which unifies twisted equivariant complex K-theory, Atiyah's `Real' K-theory, and their variants. In a general setting, we formulate this K-theory by using Fredholm operators, and establish basic properties such as the Bott periodicity and the Thom isomorphism. We also provide formulations of the K-theory based on Karoubi's gradations in both infinite and finite dimensions, clarifying their relationship with the Fredholm formulation.