A homotopy invariance theorem in coarse cohomology and k-theory

We introduce a notion of homotopy which is appropriate to the coarse geometry and topology studied by the second author in [7]. We prove the homotopy invariance of coarse cohomology, and of the if-theory of the C*-algebra associated to a coarse structure on a space. We apply our homotopy invariance results to show that if Af is a Hadamard manifold then the inverse of the exponential map at any point 0 induces an isomorphism between the K-theory groups of the C*-algebras associated to M and its tangent space at 0 (see Theorem 7.9). This result is consistent with a coarse version of the Baum-Connes conjecture.