Algebra/Topology seminar

Speaker: Matthias Kreck (Universität Bonn)

Titel: An overlooked problem: What are the Betti numbers of closed manifolds?

Abstract: This is joint work with Don Zagier. Problem: Given a sequence b_0. b_1,…, b_n of natural numbers. Is there a closed smooth manifold M with b_i(M) = b_i? For non-orientable manifolds we have a simple and complete answer, but for oriented manifolds this simple question turns out to be very hard, it will probably never be answered. We can translate it completely into algebra and number theory. This will be explained. Then I will discuss some special cases where the problem reduces to an explicit question about Bernoulli numbers. This leads to a conjecture for which we have a lot of numerical evidence - but only god knows whether it is true…..