On the Mapping class group of nontrivial $S^2$ fiber bundles

Abstract

Let $\Sigma$ be an orientbale closed surface and let $\Sigma'$ be a nonorientable closed surface. In the paper, we show that for any nontrivial orientable $S^2$ fiber bundles $X= \Sigma \ltimes S^2$ and $X' = \Sigma' \ltimes S^2$, there are surjective homomorphisms from both $MCG(X)$ and $MCG(X')$ to $\mathbb{Z}^{\infty}$. The proof is an application of generalization of Dax invariants for embedded surfaces in 4-manifolds. The property of $MCG(X)$ and $MCG(X')$ inherits from trivial fiber bundle $\Sigma \times S^2$.