2 problems

Let $F\colon X\to X'$ be a homotopy equivalence of manifolds with torsion-free fundamental groups that restricts to a diffeomorphism on the boundary. Equip the boundaries with Riem…

Let $M$ be a closed $(2k-1)$-manifold with torsion-free fundamental group, and let $\alpha$ be a unitary representation used to define the rho-invariant $\rho(M,\alpha)$. Weinberge…