Uniform homotopy invariance of Roe Index of the signature operator

TL;DR

This paper proves that the Roe index of the signature operator remains invariant under uniform homotopy equivalences between manifolds of bounded geometry, extending to equivariant cases with group actions.

Contribution

It establishes the uniform homotopy invariance of the Roe index of the signature operator for manifolds of bounded geometry, including equivariant scenarios with group actions.

Findings

Roe index is invariant under uniform homotopy equivalences.

Invariance extends to manifolds with group actions preserving bounded geometry.

Main result applies to both non-equivariant and equivariant cases.

Abstract

In this paper we study the Roe index of the signature operator of manifolds of bounded geometry. Our main result is the proof of the uniform homotopy invariance of this index. In other words we show that, given an orientation-preserving uniform homotopy equivalence $f:(M,g) \longrightarrow (N,h)$ between two oriented manifolds of bounded geometry, we have that $f_\star(Ind_{Roe}D_M) = Ind_{Roe}(D_N)$. Moreover we also show that the same result holds considering a group $\Gamma$ acting on $M$ and $N$ by isometries and assuming that $f$ is $\Gamma$-equivariant. The only assumption on the action of $\Gamma$ is that the quotients are again manifolds of bounded geometry.