Atiyah's $L^2$-Index theorem

TL;DR

This paper presents an algebraic proof of Atiyah's $L^2$-Index Theorem, relating the index of elliptic operators on closed manifolds to equivariant indices of their coverings, using group embeddings and naturality properties.

Contribution

It provides a new algebraic proof of Atiyah's $L^2$-Index Theorem, differing from the original analytic approach, by embedding groups into acyclic groups and utilizing naturality.

Findings

Algebraic proof of Atiyah's $L^2$-Index Theorem

Embedding groups into acyclic groups is effective

Natural properties of indices are crucial in the proof

Abstract

The $L^2$-Index Theorem of Atiyah \cite{atiyah} expresses the index of an elliptic operator on a closed manifold $M$ in terms of the $G$-equivariant index of some regular covering $\widetilde{M}$ of $M$, with $G$ the group of covering transformations. Atiyah's proof is analytic in nature. Our proof is algebraic and involves an embedding of a given group into an acyclic one, together with naturality properties of the indices.