Topological Obstructions to Dynamical Convexity

We study the topological obstructions of dynamical convexity on contact manifolds focusing on fillability by cotangent bundles and subcritical surgeries. Using links to algebraic geometry, we motivate and define a stronger version of dynamical convexity, and investigate the topology of these manifolds. More precisely, we show that strongly dynamically convex contact manifolds cannot arise as a unit cotangent bundle $(ST^*M,\lambda_{std})$ of a closed manifold $M$ and in particular that simply connected dynamically convex contact manifolds cannot be filled by cotangent bundles. We demonstrate that dynamical convexity can be used to recover homotopy groups of topologically simple fillings with vanishing symplectic homology. We also show obstructions to dynamical convexity that come from studying different kinds of subcritical surgeries.