Geometry and Topology of open manifolds – GTO

The topology of open manifolds is much richer than that of compact manifolds. For instance, in each dimension n, there is (up to homeomorphism) only one manifold homotopically equivalent to the n-sphere: the n-sphere itself. By contrast, there are for every n>2 uncountably many Whitehead manifolds, i.e. contractible open n-manifolds not homeomorphic to Euclidean n-space.

In addition to being interesting objects in their own right, open manifolds appear in the study of compact manifolds, for instance as limits of blow-ups of geometric flows or other special sequences of Riemannian manifolds. The open manifolds appearing in these particular contexts are expected, and sometimes known, to have well behaved topology and geometry; therefore they deserve special attention.

In this project, we propose to study the topology of open manifolds with tools coming from Riemannian geometry. This idea has been highly successful in the compact case, one of the highlights being Perelman's proof of the Poincaré et Geometrization Conjectures using Hamilton's Ricci flow We aim at broadening the scope of these techniques.

Our goal is to gather a team of world-renowned French specialists of these questions, with expertise in topology, geometry, and analysis, and to pool their abilities in order to make significant progress on this program.