Geometry and Topology Seminar

We will discuss two recent results on positive scalar curvature on circle bundles. In the context of trivial circle bundles over 4-manifolds, we consider two problems: Gromov's width inequality conjecture, and Rosenberg's S^1-stability conjecture. Both conjectures have counterexamples in dimension 4 based on Seiberg-Witten invariants. Nevertheless, we show that in the simply connected case both of these results are true upon considering 4-manifolds up to homeomorphism. We also obtain a result up to stabilization in the non-simply connected case. In the context of nontrivial bundles, we construct infinitely many examples of PSC circle bundles over enlargeable manifolds in all dimensions greater than 3. This answers a question of Gromov. We shall emphasize some analogies between symplectic geometry and positive scalar curvature that we encountered in the process. This is joint work with B. Sen.