Topological Obstructions to Dynamical Convexity

TL;DR

This paper investigates topological obstructions to dynamical convexity in contact manifolds, showing certain fillability and topological properties are incompatible with strong dynamical convexity, using tools from algebraic geometry and surgery theory.

Contribution

It introduces a stronger version of dynamical convexity, establishes new topological obstructions, and links dynamical convexity to the topology of fillings and surgeries.

Findings

Strongly dynamically convex manifolds cannot be cotangent bundles of closed manifolds.

Simply connected dynamically convex manifolds cannot be filled by cotangent bundles.

Dynamical convexity helps recover homotopy groups of certain fillings.

Abstract

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.