Event № 338
Manifolds of negative sectional curvature are an object of interest and their study goes back to Cartan and Hadamard.
It is well known that the topology of such manifolds is controlled, to some extent, by their volume. This is best illustrated in dimension 2: the homemorphism type of a compact orientable surface is determined by its volume (suitably normalized) - this follows from the celebrated Gauss-Bonnet theorem. Gromov proved in 1978 that the Betti numbers of negatively curved manifolds are bounded by means of the volume in every dimension, but also provided an example of a sequence of negatively curved 3-manifolds of uniformly bounded volume and pairwise different first integral homology. A crucial tool in Gromov's proof is the famous "thin-thick decomposition" of a manifold.
In my talk I will report on a joint work with Gelander and Sauer, in which we introduce a modification of this decomposition that gives a better model for the topology of a manifold: a negatively curved manifold is homotopic to a simplicial complex with handles, where the number of simplices is bounded by means of the volume of the manifold. This shows in particular that Gromov's 3d example could not be given in higher dimensions and that in dimension 5 and more the number of homeomorphism types of manifolds is bounded by means of the volume.