Mapping class group of $\Sigma_g \times S^1$

Question

The mapping class group of a topological space $X$ is the group of homeomorphisms $X \to X$ considered up to isotopy. If $\Sigma_g$ is the closed orientable surface of genus $g$, what is known about the mapping class group of $\Sigma_g \times S^1$?

Answer 1

As $\Sigma_g \times S^1$ is a surface bundle over the circle it is a Haken 3-manifold. Due to Waldhausen in On Irreducible 3-manifolds Which are Sufficiently Large (Haken 3-manifolds) we have the natural homomorphism $\text{Mod}(\Sigma_g \times S^1) \rightarrow \text{Out}(\pi_1(\Sigma_g \times S^1))$ is an isomorphism. Here $\text{Mod}(M) \approx \text{Homeo}(M)/ \text{Homeo}_0(M)$ the mapping class group.