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

Abstract

Let $\Sigma$ be an orientbale closed surface and let ${\Sigma }^{\prime }$ be a nonorientable closed surface. In the paper, we show that for any nontrivial orientable $S^2$ fiber bundles $X=\Sigma ⋉S^2$ and $X^{\prime }={\Sigma }^{\prime }⋉S^2$, there are surjective homomorphisms from both $MCG(X)$ and $MCG(X^{\prime })$ 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 $MCG(X^{\prime })$ inherits from trivial fiber bundle $\Sigma \times S^2$.

14 pages