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.