On the Mapping class group of nontrivial S² fiber bundles

2025/09/20 by Huizheng Guo, Guo, Huizheng
Mathematics · #57R50 #57R52 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2509.16520

openalex publication_date 2025/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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\ltimes S² and X' = Σ' \ltimes S², there are surjective homomorphisms from both MCG(X) and MCG(X') to ℤ^∞. 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 Σ× S².

Citations

• Mapping class groups of simply connected 4-manifolds with boundary
• Knotted 3-balls in S⁴

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