## SPESSATO STEFANO

PhD Graduate**PhD program:**: XXXIII

**supervisor**: G. Mondello

**advisor**: P. Piazza

**Thesis title:**Lipschitz-homotopy invariants: L^2-cohomology, Roe index and ρ-class

We study Lipschitz-homotopy equivalences between manifolds of bounded geometry. In particular, we suppose the Lipschitz-homotopy equivalences are G-equivariant with respect to the action of a group of isometries G. In this Thesis we prove that reduced L2-cohomology, un-reduced L2-cohomology, and the Roe index of the signature operator are Lipschitz-homotopy invariants. We conclude this Thesis by defining for each Lipschitz-homotopy equivalence a rho class that only depends on the Lipschitz-homotopy class of the homotopy equivalence.