$L^2$-cohomology and quasi-isometries on the ends of unbounded geometry

Abstract

The paper explores the minimal and maximal $L^2$-cohomology of oriented Riemannian manifolds, focusing on both the reduced and the unreduced versions. The main result is the proof of the invariance of the $L^2$-cohomology groups under uniform homotopy equivalences that are quasi-isometric on the unbounded ends. A uniform map is a uniformly continuous map such that the diameter of the preimage of a subset is bounded in terms of the diameter of the subset itself. Moreover, a map $f$ between two Riemannian manifolds $(X,g)$ and $(Y,h)$ is quasi-isometric on the unbounded ends if $X=M\cup E_X$, where $M$ is the interior of a manifold of bounded geometry with boundary, $E_X$ is an open subset of $X$ and the restriction of $f$ to $E_X$ is a quasi-isometry. Finally, some consequences are shown: the main ones are the definition of a mapping cone for $L^2$-cohomology and the invariance of the $L^2$-signature.