## Graduate Studies

## First Advisor

Alexander Zupan

## Degree Name

Doctor of Philosophy (Ph.D.)

## Department

Mathematics

## Date of this Version

7-2024

## Document Type

Dissertation

## Citation

A dissertation presented to the faculty of the Graduate College of the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy

Major: Mathematics

Under the supervision of Professor Alexander Zupan

Lincoln, Nebraska, July 2024

## Abstract

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} × *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 Σ

_{g,1}-bundle over the torus, and let

*W*′ be defined similarly. If

*f*: ∂

*W*′ → ∂

*W*is an orientation-preserving diffeomorphism of the

*T*

^{3}-boundary, we have that

*X*=

*W*∪

*′ fibers over the circle.*

_{f}WWe 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* = ∂*Y* and that *f* : *F* × *S*^{1} → *F* × *S*^{1} is an orientation- preserving diffeomorphism. Then, we show that the *f*-spin of *Y* admits a (2*g* − *h*; *g*) 4-section if *h* ≠ 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.

Advisor: Alexander Zupan

## Recommended Citation

Meyer, Nicholas Paul, "Torus Surgery, Fibrations, Multisections, and Spun 4-Manifolds" (2024). *Dissertations and Doctoral Documents from University of Nebraska-Lincoln, 2023–*. 151.

https://digitalcommons.unl.edu/dissunl/151

## Comments

Copyright 2024, Nicholas Paul Meyer. Used by permission