What are the Betti numbers of closed manifolds?

Matthias Kreck, Univsersität Bonn

Online Talk 

I will report about  joint work with Don Zagier. We study the following obvious question: Given natural numbers b_0, b_1,…., b_n. Is there a closed smooth manifold M with b_i(M) = b_i? Since we only look at connected manifolds b_0 = 1. Then b_n decides whether M is orientable or not. In the non-oriented case there is a simple complete answer. But the oriented case is very mysterious. We can translate the problem completely into algebra and number theory but this doesn’t mean that the question is answered. We have partial results, for example for which even n does there exist a closed oriented manifold with all Betti numbers below the middle dimension 0 (except b_0) and b_{n/2} an odd number. This and some rather curious results will be explained.