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

Suppose that $W$ and $W^{\mathrm{\prime }}$ 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 ${\mathrm{\Sigma }}_{g,1}$ bundles over the $2$-torus with monodromy fixing the boundary of the fiber pointwise. If $f:\mathrm{\partial }{W}^{\mathrm{\prime }}\to \mathrm{\partial }W$ is an orientation-preserving diffeomorphism of the $3$-torus boundaries, we have that $X=W{\cup }_f{W}^{\mathrm{\prime }}$ is a closed, oriented $4$-manifold that fibers over $S^1$. In particular, if $W^{\mathrm{\prime }}=T^2\times D^2$ 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.