Extending Fibrations of the $3$-Torus and Applications to Torus Surgery in $4$-Manifolds

TL;DR

This paper investigates how certain 4-manifolds that fiber over the circle can be constructed via gluing along torus boundaries, with applications to torus surgery in 4-manifolds, including explicit results for surgeries on unknots in S^1×S^3.

Contribution

It extends fibrations of 4-manifolds with torus boundary components and applies these results to analyze torus surgeries, including new explicit diffeomorphism classifications.

Findings

Torus surgeries in S^1×Y produce 4-manifolds that fiber over S^1.

Torus surgery along S^1×U in S^1×S^3 yields S^1× lens space.

Extension of fibrations to manifolds obtained by gluing along torus boundaries.

Abstract

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 $\Sigma_{g,1}$ 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.