Dissertation
Trisections of 4-manifolds
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2021
DOI: 10.17077/etd.006116
Abstract
This thesis focuses on understanding the theory of trisections of 4-dimensional spaces and its relationship with problems in dimension three. A trisection is a decomposition of 4-dimensional space into three standard pieces. Trisections are an exciting way of thinking about the fourth dimension since they bring results and techniques of dimensions two and three to the wild world of 4-dimensional space. The theory of trisections is a new and active area of research in modern mathematics. Mathematicians have worked to expand the theory, compute examples and invariants, and use it to study 4-dimensional geometry. We study trisections from three perspectives: using knots and links, motivated from classical techniques of 3-manifold topology, and studying their diagrams.
One can extract a non-negative integer from a trisected 4-dimensional space called the ``trisection genus." Its understanding would lead us to answer one of the most challenging open problems in topology: the 4-dimensional smooth Poincar\'e Conjecture. One of this thesis's main results is constructing an infinite family of distinct trisections of minimal-genus. This result is an interesting example of the interplay between dimensions three and four since the proof technique uses the knot theory machinery.
One of the core techniques in modern low-dimensional topology is thin position for 3-dimensional spaces, introduced by Gabai for knots, further developed by Scharlemann and Thompson in the 90s, and extended in more general settings by Tomova and others. One of this thesis's main contributions is introducing a notion of thin position in dimension four. Chapter 2 mixes ideas of the theory of trisections with the original work of Scharlemann and Thompson.
In practice, one can use diagrams to describe trisections: particular drawings on a surface. In general, it is not evident whether two trisection diagrams represent the same decomposition or how the diagram of a given trisection looks. This thesis describes the diagrammatics of $\star$-trisections, which increase the allowed diagrams and permit us to address those problems in new families of examples. Most of the work in Chapter 3 describes concrete examples of using $\star$-trisections to draw trisections diagrams for several spaces.
Details
- Title: Subtitle
- Trisections of 4-manifolds
- Creators
- José Román Aranda Cuevas
- Contributors
- Maggy Tomova (Advisor)Charles Frohman (Committee Member)Keiko Kawamuro (Committee Member)Benjamin Cooper (Committee Member)Mohammad Tehrani (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2021
- DOI
- 10.17077/etd.006116
- Publisher
- University of Iowa
- Number of pages
- xiii, 146 pages
- Copyright
- Copyright 2021 José Román Aranda Cuevas
- Comment
- This thesis has been optimized for improved web viewing. If you require the original version, contact the University Archives at the University of Iowa: https://www.lib.uiowa.edu/sc/contact/
- Language
- English
- Description illustrations
- color illustrations
- Description bibliographic
- Includes bibliographical references (pages 142-146).
- Public Abstract (ETD)
This thesis focuses on understanding the theory of trisections of 4-dimensional spaces and its relationship with problems in dimension three. A trisection is a decomposition of 4-dimensional space into three standard pieces. Trisections are an exciting way of thinking about the fourth dimension since they bring results and techniques of dimensions two and three to the wild world of 4-dimensional space. The theory of trisections is a new and active area of research in modern mathematics. Mathematicians have worked to expand the theory, compute examples and invariants, and use it to study 4-dimensional geometry. We study trisections from three perspectives: using knots and links, motivated from classical techniques of 3-manifold topology, and studying their diagrams.
- Academic Unit
- Mathematics
- Record Identifier
- 9984097076602771
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