Extending Fibrations of the $3$-Torus and Applications to Torus Surgery in $4$-Manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909971044630528 |
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| author | Meyer, Nicholas |
| author_facet | Meyer, Nicholas |
| contents | Suppose that $W$ and $W'$ are smooth, compact, and oriented $4$-manifolds that are either diffeomorphic to $S^1$ times the exterior $E_Y(K)$ of a fibered knot $K$ in a closed, connected, orientable $3$-manifold $Y$, or are diffeomorphic to $Σ_{g,1}$ bundles over the $2$-torus with monodromy fixing the boundary of the fiber pointwise. If $f: \partial W' \to \partial W$ is an orientation-preserving diffeomorphism of the $3$-torus boundaries, we have that $X = W \cup_f W'$ is a closed, oriented $4$-manifold that fibers over $S^1$. In particular, if $W' = T^2 \times D^2$ and $W = S^1 \times E_Y(K)$, then our result shows that the result of doing torus surgery in $S^1\times Y$ along $S^1 \times K$ is a $4$-manifold that fibers over $S^1$. Furthermore, we extend work of Zentner by showing that the result of torus surgery along $S^1$ times the unknot $\mathcal{U}$ in $S^1 \times S^3$ is diffeomorphic to $S^1$ times a lens space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_17595 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extending Fibrations of the $3$-Torus and Applications to Torus Surgery in $4$-Manifolds Meyer, Nicholas Geometric Topology 57K40 (Primary) 57R22 (Secondary) Suppose that $W$ and $W'$ are smooth, compact, and oriented $4$-manifolds that are either diffeomorphic to $S^1$ times the exterior $E_Y(K)$ of a fibered knot $K$ in a closed, connected, orientable $3$-manifold $Y$, or are diffeomorphic to $Σ_{g,1}$ bundles over the $2$-torus with monodromy fixing the boundary of the fiber pointwise. If $f: \partial W' \to \partial W$ is an orientation-preserving diffeomorphism of the $3$-torus boundaries, we have that $X = W \cup_f W'$ is a closed, oriented $4$-manifold that fibers over $S^1$. In particular, if $W' = T^2 \times D^2$ and $W = S^1 \times E_Y(K)$, then our result shows that the result of doing torus surgery in $S^1\times Y$ along $S^1 \times K$ is a $4$-manifold that fibers over $S^1$. Furthermore, we extend work of Zentner by showing that the result of torus surgery along $S^1$ times the unknot $\mathcal{U}$ in $S^1 \times S^3$ is diffeomorphic to $S^1$ times a lens space. |
| title | Extending Fibrations of the $3$-Torus and Applications to Torus Surgery in $4$-Manifolds |
| topic | Geometric Topology 57K40 (Primary) 57R22 (Secondary) |
| url | https://arxiv.org/abs/2512.17595 |