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October 3, 2025Open Access

On the Mapping class group of nontrivial S² fiber bundles

Key Points

Surjective homomorphisms exist from both MCG_0(X) and MCG_0(X') to Z^∞, revealing their complexity.
The study applies a generalization of Dax invariants for embedded surfaces in 4-manifolds to yield results.
The properties of MCG_0(X) and MCG_0(X') inherit from a trivial fiber bundle Σ x S^2, suggesting foundational connections.
These findings expand our understanding of fiber bundles and their mapping class groups in topology.

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 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.

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Huizheng Guo (Sat,) studied this question.

synapsesocial.com/papers/68e040f3a99c246f578b3711 — DOI: https://doi.org/10.48550/arxiv.2509.16520

On the Mapping class group of nontrivial S² fiber bundles

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