4-manifolds with a given boundary
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914106449068032 |
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| author | Conway, Anthony Kasprowski, Daniel |
| author_facet | Conway, Anthony Kasprowski, Daniel |
| contents | This paper studies the homotopy and homeomorphism classifications of $4$-manifolds with boundary. Given $4$-manifolds $X_0$ and $X_1$ with fundamental group $π$, we consider the problem of extending a homotopy equivalence $h \colon \partial X_0 \to \partial X_1$ to a homotopy equivalence $X_0 \to X_1$. We solve this problem in broad settings for a class of groups that includes free groups, finite cyclic groups, finite dihedral groups, solvable Baumslag-Solitar groups, and many $3$-manifold groups. When the fundamental group is additionally assumed to be good, we use surgery theory to list situations when a homeomorphism $h\colon\partial X_0 \to\partial X_1$ extends to a homeomorphism $X_0 \to X_1$. The outcome recovers results of Boyer in the simply-connected case and work of the first author and Powell when $π\cong \mathbb{Z}$ and the $\partial X_i$ have torsion Alexander module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18836 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 4-manifolds with a given boundary Conway, Anthony Kasprowski, Daniel Geometric Topology This paper studies the homotopy and homeomorphism classifications of $4$-manifolds with boundary. Given $4$-manifolds $X_0$ and $X_1$ with fundamental group $π$, we consider the problem of extending a homotopy equivalence $h \colon \partial X_0 \to \partial X_1$ to a homotopy equivalence $X_0 \to X_1$. We solve this problem in broad settings for a class of groups that includes free groups, finite cyclic groups, finite dihedral groups, solvable Baumslag-Solitar groups, and many $3$-manifold groups. When the fundamental group is additionally assumed to be good, we use surgery theory to list situations when a homeomorphism $h\colon\partial X_0 \to\partial X_1$ extends to a homeomorphism $X_0 \to X_1$. The outcome recovers results of Boyer in the simply-connected case and work of the first author and Powell when $π\cong \mathbb{Z}$ and the $\partial X_i$ have torsion Alexander module. |
| title | 4-manifolds with a given boundary |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2510.18836 |