Algebra/Topology seminar

Fabian Hebestreit, Tautological classes of aspherical manifolds

Abstract: Joint with M.Land, W.Lück and O.Randal-Williams

Tautological classes, or generalised Morita-Miller-Mumford classes as they are often called, are characteristic classes of manifold bundles derived from tangential data. They have recently come to the foreground through the work of Madsen, Tillmann and Weiss on surface bundles, and then Galatius and Randal-Williams for higher dimensional fibres. Their results describe the cohomology of the diffeomorphism group in degrees linearly bounded by the genus of the manifold in terms of these classes and close relatives. 

Aspherical manifolds are an interesting class of manifold that have vanishing genus and indeed, I will speak about a result of ours that shows the vanishing of tautological classes on them away from two obstructions: The consequences of the surface case and hyperbolisation.