David Wraith, National University of Ireland

Title: Intermediate curvatures and Gromov's Betti number bound.

Abstract: We study curvatures which interpolate between positive Ricci curvature and positive sectional curvature. Gromov's upper bound on the total Betti number in the presence of a lower sectional curvature bound was shown to fail for positive Ricci curvature by Sha and Yang. We show that the total Betti number bound also fails for a range of intermediate curvatures, from positive Ricci curvature up to roughly halfway towards positive sectional curvature. Like Sha and Yang, we approach the problem via surgery, and establish the first surgery results for these curvature conditions. The difficulty to overcome is algebraic, and we will discuss this issue and its resolution. This is joint work with Philipp Reiser.