Dynamical and cohomological obstructions to extending group actions

TL;DR

This paper investigates cohomological and dynamical obstructions to extending boundary group actions of 3-manifolds to the interior, showing that certain boundary actions cannot be extended to the entire manifold.

Contribution

It introduces new cohomological criteria that prevent extending boundary group actions to the interior of 3-manifolds, especially for torus boundaries.

Findings

Torus actions on boundary do not extend to the interior for non-solid torus 3-manifolds.

Cohomological obstructions are key in understanding extension problems.

Results apply to manifolds with torus or sphere boundaries.

Abstract

We study cohomological obstructions to extending group actions on the boundary $\partial M$ of a $3$-manifold to a $C^0$-action on $M$ when $\partial M$ is diffeomorphic to a torus or a sphere. In particular, we show that for a $3$-manifold $M$ with torus boundary which is not diffeomorphic to a solid torus, the torus action on the boundary does not extend to a $C^0$-action on $M$.