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

Abstract: Suppose that $W$ and $W&#39;$ are smooth, compact, and oriented $4$-manifolds that are either diffeomorphic to $S<sup>1$ 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 $f: \partial W&#39; \to \partial W$ is an orientation-preserving diffeomorphism of the $3$-torus boundaries, we have that $X = W \cup_f W&#39;$ is a closed, oriented $4$-manifold that fibers over $S<sup>1$. In particular, if $W&#39; = T² \times D<sup>2$ and $W = S¹ \times E_Y(K)$, then our result shows that the result of doing torus surgery in $S^1\times Y$ along $S¹ \times K$ is a $4$-manifold that fibers over $S<sup>1$. Furthermore, we extend work of Zentner by showing that the result of torus surgery along $S<sup>1$ times the unknot $\mathcal{U}$ in $S¹ \times S<sup>3$ is diffeomorphic to $S<sup>1$ times a lens space.