Lipschitz-homotopy invariants:$L^2$-cohomology, Roe index and $\rho$-class

TL;DR

This thesis establishes Lipschitz-homotopy invariants for manifolds of bounded geometry, including $L^2$-cohomology, Roe index, and a $ ho$-class, demonstrating their invariance under Lipschitz-homotopy equivalences.

Contribution

It introduces a functor capturing Lipschitz-homotopy invariants and proves invariance of $L^2$-cohomology, Roe index, and defines a $ ho$-class in K-theory related to Lipschitz-homotopy.

Findings

$L^2$-cohomology is Lipschitz-homotopy invariant.

Roe index of the signature operator is invariant under Lipschitz-homotopy.

Defined a $ ho$-class in K-theory depending only on the Lipschitz-homotopy class.

Abstract

In this thesis we talk about lipschitz-homotopy invariants for manifolds of bounded geometry. There are three main results: the first one is the definition of a controvariant functor between the category of manifolds of bounded geometry with uniformly proper lipschitz maps and the category of complex vector space together linear maps. As consequence of this we obtain that the (un)-reduced $L^2$-cohomology is a lipschitz-homotopy invariant. The second result is the invariance of the Roe index of the signature operator under lipschitz-homotopy equivalence which preserves the orientations. Finally, the last result is the following: given a lipschitz-homotopy equivalence wich preserves the orientations between manifold of bounded geometry, we define a $\rho$-class in the K-theory group of the structure algebra of the codomain which is related to the lipschitz-homotopy. This class only depends on the domain and on the lipschitz-homotopy class of the map.