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The cohomology and homotopy groups of moduli spaces of smooth manifolds are some of the most basic invariants of manifold bundles: the former contain all the characteristic classes and the latter classify smooth bundles over spheres. Complete calculations of these groups are challenging, even for the simplest compact manifolds. It is then desirable to know, at least, some qualitative information, for example whether these groups are (degreewise) finitely generated.
In this talk, I will discuss a method to attack this question which leads to the following theorem: if M is a closed smooth manifold of even dimension > 5 with finite fundamental group, then the cohomology and higher homotopy groups of BDiff(M) are finitely generated abelian groups.
This is joint work with M. Krannich and A. Kupers. |