Torus Surgery, Fibrations, Multisections, and Spun 4-Manifolds

Authors

Nicholas Paul Meyer, University of Nebraska-LincolnFollow

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

Comments

Copyright 2024, Nicholas Paul Meyer. Used by permission

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 S¹ × 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 T³-boundary, we have that X = W ∪_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 = ∂Y and that f : F × S¹ → F × S¹ is an orientation- preserving diffeomorphism. Then, we show that the f-spin of Y admits a (2g − 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