# An infinite-dimensional phenomenon in finite-dimensional metric topology

# An infinite-dimensional phenomenon in finite-dimensional metric topology

We introduce a notion of “deformation equivalence” for topological manifolds. Deformation equivalent manifolds are homotopy equivalent via a simple-homotopy equivalence inducing isomorphisms on rational Pontrjagin classes, but they need not be homeomorphic. If M^m, m > 6, is a closed simply connected manifold such that pi_2(M) vanishes, then there are manifolds which are deformation equivalent to M if and only if KO_m(M) has odd torsion. Indeed, for each odd torsion class tau in KO_m(M) there is a unique homotopy equivalence f:N -> M which is realized by a deformation and whose signature operator differs from that of M by tau. This gives many simply-connected examples -- for instance between S^3-bundles over S^4. If M^m, m>6, is a closed aspherical manifold, then any closed manifold deformation equivalent to M is homeomorphic to M. This is joint work with Alexander Dranishnikov and Shmuel Weinberger.