Large Riemannian manifolds which are flexible

Abstract

For each $k \in \mathbb{Z}$, we construct a uniformly contractible metric on Euclidean space which is not mod $k$ hypereuclidean. We also construct a pair of uniformly contractible Riemannian metrics on $\mathbb{R}^n$, $n \ge 11$, so that the resulting manifolds $Z$ and $Z’$ are bounded homotopy equivalent by a homotopy equivalence which is not boundedly close to a homeomorphism. We show that for these spaces the $C^*$-algebra assembly map $K_*^{lf}(Z) \to K_*(C^*(Z))$ from locally finite $K$-homology to the $K$-theory of the bounded propagation algebra is not a monomorphism. This shows that an integral version of the coarse Novikov conjecture fails for real operator algebras. If we allow a single cone-like singularity, a similar construction yields a counterexample for complex $C^*$-algebras.