The Completion Theorem in twisted equivariant $K$-Theory for proper and discrete actions

TL;DR

This paper establishes a completion theorem in twisted equivariant K-Theory for proper actions of discrete groups, extending classical results and providing new algebraic tools for understanding twisted K-theoretic invariants.

Contribution

It introduces a module structure over untwisted equivariant K-Theory and proves a completion theorem of Atiyah-Segal type for twisted equivariant K-Theory, along with a cocompletion theorem for twisted Borel K-Homology.

Findings

Proved a completion theorem for twisted equivariant K-Theory.

Constructed a module structure over untwisted equivariant K-Theory.

Established a cocompletion theorem for twisted Borel K-Homology.

Abstract

We compare different algebraic structures in twisted equivariant K-Theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-Theory, we prove a completion Theorem of Atiyah-Segal type for twisted equivariant K-Theory. Using a Universal coefficient Theorem, we prove a cocompletion Theorem for Twisted Borel K-Homology for discrete Groups.