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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.08847 |
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| _version_ | 1866910999900061696 |
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| author | Lin, Jianfeng Wang, Zhongzi |
| author_facet | Lin, Jianfeng Wang, Zhongzi |
| contents | Suppose a closed oriented $n$-manifold $M$ bounds an oriented $(n+1)$-manifold. It is known that $M$ $π_1$-injectively bounds an oriented $(n+1)$-manifold $W$. We prove that $π_1(W)$ can be residually finite if $π_1(M)$ is, and $π_1(W)$ can be finite if $π_1(M)$ is. In particular, each closed 3-manifold $M$ $π_1$-injectively bounds a 4-manifold with residually finite $π_1$, and bounds a 4-manifold with finite $π_1$ if $π_1(M)$ is finite. Applications to 3- and 4-manifolds are given:
(1) We study finite group actions on closed 4-manifolds and $π_1$-isomorphic cobordism of 3-dimensional lens spaces. Results including: (a) Two lens spaces are $π_1$-isomorphic cobordant if and only if there is a degree one map between them. (b) Each spherical 3-manifold $M\ne S^3$ can be realized as the unique non-free orbit type for a finite group action on a closed 4-manifold.
(2) The minimal bounding index $O_b(M)$ for closed 3-manifolds $M$ are defined, %and bounding Euler charicteristic $χ_b(M)$. the relations between finiteness of $O_b(M)$ and virtual achirality of aspherical (hyperbolic) $M$ are addressed. We calculate $O_b(M)$ for some lens spaces $M$. Each prime is realized as a minimal bounding index.
(3) We also discuss some concrete examples:Surface bundle often bound surface bundles, and prime 3-manifolds often virtually bound surface bundles, $W$ bounded by some lens spaces realizing $O_b$ is constructed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08847 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $π_1$-injective bounding and application to 3- and 4-manifolds Lin, Jianfeng Wang, Zhongzi Geometric Topology Suppose a closed oriented $n$-manifold $M$ bounds an oriented $(n+1)$-manifold. It is known that $M$ $π_1$-injectively bounds an oriented $(n+1)$-manifold $W$. We prove that $π_1(W)$ can be residually finite if $π_1(M)$ is, and $π_1(W)$ can be finite if $π_1(M)$ is. In particular, each closed 3-manifold $M$ $π_1$-injectively bounds a 4-manifold with residually finite $π_1$, and bounds a 4-manifold with finite $π_1$ if $π_1(M)$ is finite. Applications to 3- and 4-manifolds are given: (1) We study finite group actions on closed 4-manifolds and $π_1$-isomorphic cobordism of 3-dimensional lens spaces. Results including: (a) Two lens spaces are $π_1$-isomorphic cobordant if and only if there is a degree one map between them. (b) Each spherical 3-manifold $M\ne S^3$ can be realized as the unique non-free orbit type for a finite group action on a closed 4-manifold. (2) The minimal bounding index $O_b(M)$ for closed 3-manifolds $M$ are defined, %and bounding Euler charicteristic $χ_b(M)$. the relations between finiteness of $O_b(M)$ and virtual achirality of aspherical (hyperbolic) $M$ are addressed. We calculate $O_b(M)$ for some lens spaces $M$. Each prime is realized as a minimal bounding index. (3) We also discuss some concrete examples:Surface bundle often bound surface bundles, and prime 3-manifolds often virtually bound surface bundles, $W$ bounded by some lens spaces realizing $O_b$ is constructed. |
| title | $π_1$-injective bounding and application to 3- and 4-manifolds |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2506.08847 |