$L^{2}$-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.