Dynamical and cohomological obstructions to extending group actions

We study cohomological obstructions to extending group actions on the boundary $\mathrm{\partial }M$ of a $3$-manifold to a $C^0$-action on $M$ when $\partial M$isdiffeomorphictoatorusorasphere\mathrm{.}Inparticular,weshowthatfora$3$-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 $M$.