Braiding non-ribbon surfaces and constructing broken fibrations on smooth 4-manifolds

Authors

Mark Clifford Hughes

Type

Text

Type

Dissertation

Advisor

Sullivan, Dennis | Viro, Oleg Y | Plamenevskaya, Olga | Rocek, Martin.

Date

2014-12-01

Keywords

Mathematics | 4-manifolds, geometric topology, knot theory, Lefschetz fibrations

Department

Department of Mathematics.

Language

en_US

Source

This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.

Identifier

http://hdl.handle.net/11401/76398

Publisher

The Graduate School, Stony Brook University: Stony Brook, NY.

Format

application/pdf

Abstract

In this thesis we study various notions of surface braidings in 4-space, and their applications to the construction of singular fibrations on smooth oriented 4-manifolds. We define the notion of braided link cobordisms in S3 × [0,1], which generalize Viro's closed 2-braids in S4. We prove that via isotopy any properly embedded oriented surface W in S3 × [0,1] can be brought to this special position, and that the isotopy can be taken rel boundary when the boundary already consists of closed braids. These surfaces are closely related to another notion of surface braiding in D2 × D2, called braided surfaces with caps, which generalize Rudolph's braided surfaces. We use these to construct broken Lefschetz fibrations on smooth 4-manifolds. We first consider the case when the 4-manifold X has connected non-empty boundary, and construct the desired fibration as the composition of a covering X &rarr D2 × D2 branched along a singular braided surface with caps, with the projection map pr₂: D2 × D2 &rarr D2. Proceeding in this way gives us the ability to specify the behavior of our fibration along the boundary of X. Broken Lefschetz fibrations on closed manifolds are then obtained by combining this result with a construction of Gay and Kirby. This allows us to reprove earlier existence results due to Akbulut and Karakurt, Baykur, and Lekili, giving a more concrete geometric approach to constructing these fibrations. | 92 pages

Recommended Citation

Hughes, Mark Clifford, "Braiding non-ribbon surfaces and constructing broken fibrations on smooth 4-manifolds" (2014). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2321.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2321