On the Mapping class group of nontrivial S² fiber bundles

Let $Σ$ be an orientbale closed surface and let $Σ'$ be a nonorientable closed surface. In the paper, we show that for any nontrivial orientable S² fiber bundles X= Σ S² and X' = Σ' S², there are surjective homomorphisms from both MCG₀(X) and MCG₀(X') to Z∞. The proof is an application of generalization of Dax invariants for embedded surfaces in 4-manifolds. The property of MCG₀(X) and MCG₀(X') inherits from trivial fiber bundle Σ× S².