# Relative kappa-classes

HHHW04 - Manifolds

Diff(Dn), the the space of diffeomorphisms of the n-disc fixed near the boundary has rich rational topology. For example, Weiss's discovery of surreal'' Pontrjagin classes leads to the existence of rationally non-trivial homotopy classes in BDiff(Dn).

For any smooth n-manifold M, extension by the identity induces a map BDiff(D^n) \to BDiff(M). In this talk I will report on joint work with Wolfgang Steimle and Thomas Schick, where we consider the problem of computing the image of the Weiss classes'' under the maps on homotopy and homology induced by extension. This problem naturally leads one to consider relative kappa-classes.

Via relative kappa-classes, we show that the maps induced by extension are rationally non-trivial for a wide class of manifolds M, including aspherical manifolds (homology, hence also homotopy) and stably parallelisable manifolds (homotopy). When M is aspherical, our arguments rely on vanishing results for kappa-classes due to Hebestreit, Land, Lueck and Randal-Williams.

This talk is part of the Isaac Newton Institute Seminar Series series.