Freed–Moore $K$-theory

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.

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Received 5 October 2019

Accepted 3 February 2021

Published 16 July 2024