A compact $n$-manifold $X$ is fibered if it is a fiber bundle where the fiber $F$ and base space $B$ are manifolds. Fibered manifolds are particularly nice, as they are essentially classified by their monodromy maps. Two common examples of 4-dimensional fibered manifolds are surface bundles over surfaces and 3-manifold bundles over the circle.

The main focus of this dissertation is to investigate fibered 4-manifolds whose boundaries are the 3-torus and how these manifolds glue together to give new closed, fibered 4-manifolds. In particular, suppose W is diffeomorphic to $S^1\times E_Y(K)$ where $Y$ is a closed, oriented 3-manifold and $K$ is a fibered knot in $Y$, or that $W$ is diffeomorphic to a $\Sigma_{g,1}$-bundle over the torus, and let $W′$ be defined similarly. If $f:\partial W’ \to \partial W$ is an orientation-preserving diffeomorphism of the $T^3$ -boundary, we have that $X = W \cup_f W’$ fibers over the circle. We also study spun 4-manifolds and construct 4-secting Morse 2-functions on these manifolds. Suppose that $Y$ is a compact, oriented, connected 3-manifold with connected boundary $F = \partial Y$ and that $f:F \times S^1 \to F \times S^1$ is an orientation-preserving diffeomorphism. Then, we show that the $f$-spin of $Y$ admits a $(2g - h; g)$ 4-section if $h \neq -1$ or if $h = 1$ and $f$ is isotopic to the identity, where $h$ is the genus of $F$ and $g$ is the Heegaard genus of $Y$. This generalizes the work of Meier on trisections of spun 4-manifolds and of Kegel and Schmäschke on trisections of $4$-dimensional open book decompositions.

We define a pants distance for knotted surfaces in 4-manifolds which generalizes the complexity studied by Blair-Campisi-Taylor-Tomova for surfaces in the 4-sphere. We determine that if the distance computed on a given diagram does not surpass a theoretical bound in terms of the multisection genus, then the (4-manifold, surface) pair has a simple topology. Furthermore, we calculate the exact values of our invariants for many new examples such as the spun lens spaces. We provide a characterization of genus two quadrisections with distance at most six.

Meier and Zupan proved that an orientable surface $\mathcal{K}$ in $S^4$ admits a tri-plane diagram with zero crossings if and only if $\mathcal{K}$ is unknotted. We determine the minimal crossing numbers of nonorientable unknotted surfaces in $S^4$, proving that $c(\mathcal{P}^{n,m}) = \max\{1,|n-m|\}$, where $\mathcal{P}^{n,m}$ denotes the connected sum of $n$ unknotted projective planes with normal Euler number $+2$ and $m$ unknotted projective planes with normal Euler number $-2$. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers.