Dynamical and cohomological obstructions to extending group actions

Abstract

Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary $$\partial M$$ of a 3-manifold to a $$C^0$$ -action on M. Among other results, we show that for a 3-manifold M, the $$S^1 \times S^1$$ action on the boundary does not extend to a $$C^0$$ -action of $$S^1 \times S^1$$ via homeomorphisms that are isotopic to the identity as a discrete group on M, except in the trivial case $$M \cong D^2 \times S^1$$ .