Every $S^2$-Bundle over a positive genus($g>1$) Surface has Mapping Class Group of Infinite Rank

Abstract

In 2025, Lin, Wu, Xie and Zhang extend Dax invariant from embedded disks to embedded surfaces and show that there is surjective homomorphism from $MCG(M)$ to ${\mathbb{Z}}^{\infty }$ for $M=\Sigma \times S^2$. Let $\Sigma$ be a closed surface with positive$(g>1)$ genus, in the paper we show that for any nontrivial $S^2$ fiber bundles $X=\Sigma ⋉S^2$, whether $X$ orientable or nonorientable, there is a surjective homomorphism from $MCG(X)$ to ${\mathbb{Z}}^{\infty }$. The proof is an application of generalization of Dax invariants for embedded surfaces and the restriction on pseudo-isotopy subgroup $MC{G}_0(X)$ also surjects on ${\mathbb{Z}}^{\infty }$. The property of $MCG(X)$ inherits from trivial fiber bundle $\Sigma \times S^2$ and main theorem shows that the algebraic size of the smooth mapping class group of $S^2$-fiber bundles does not depend on the triviality of the bundle.

arXiv comment15 pages