Bonn Topology Group - Abstracts

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16 January 2018
MANUEL KRANNICH (University of Copenhagen): On characteristic classes of exotic manifold bundles


The well-studied inertia group of a closed manifold M captures the behavior of the diffeomorphism type of M when changing the smooth structure of M on a codimension zero disc, that is, when taking the connected sum M#S with an exotic sphere S. Instead of comparing their diffeomorphism types, one might try to compare the group of diffeomorphisms of M and M#S. However, the space of diffeomorphisms Diff(M) can be shown to naturally decompose as a twisted product whose factors are, up to homotopy, insensitive to replacing M with M#S. This suggests that it is hard to detect the possible exotic nature of M#S by means of homotopical properties of its group of diffeomorphisms. Using recent progress in manifold theory by Galatius and Randal-Williams, together with computations in stable homotopy theory, I will discuss results on the behavior of the cohomology of BDiff(M) when replacing M with M#S.

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