## Citation

Choi, Yanglim
(1998)
*(3,1)-Surfaces via Branched Surfaces.*
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/yw04-qx32.
https://resolver.caltech.edu/CaltechTHESIS:11202019-173910635

## Abstract

Loosely speaking, a (*n*,1)-surface is a very nicely immersed π₁-injective surface in a 3-manifold. Its concept was born around 1981 by Peter Scott in his work on Seifet fibered spaces. It has been shown that if a 3-manifold *M* contains a (4,1)-surface, then its universal cover is R³ and π₁(*M*) determines *M* up to homeomorphism. Homotopic homeomorphisms are isotopic on a 3-manifold containing a (3,1)-surface. On the other hand, some class of 3-manifolds, such as manifolds with nonpositive cubing, by Aitchison and Rubinstein, are known to contain (4,1)-surfaces. One natural question, then, is how 'big' is the set of 3-manifolds with (4,1)-surfaces in the set of all 3-manifolds. Similar question for embedded π₁-injective surfaces, called *incompressible* surfaces, has been answered in a work of Floyd and Oertel around 1980. They showed th a t the set of incompressible surfaces in a 3-manifold is carried by a finite number of branched surfaces. Combining this with a theorem of Hatcher, one can reasonably argue that 3-manifolds containing incompressible surfaces, called Haken manifolds, are scarce. In this paper we prove a similar result in the context of (3,1)-surfaces and non Haken 3-manifolds.

**Theorem 1** *If M is a non Haken 3-manifold, then the set o f (3,1)-surfaces in M are embeddedly carried by a finite number of branched surfaces.*

'Embeddedly carried' is a precise generalization of 'carried' in the context of immersed surfaces. Careful examination of when the theorem is not true will lead one to obtain a sequence of least area embedded disks in *M* that limits to an essential measured lamination of *M*. Such lamination always approximates an incompressible surface in M . In some cases euler characteristic of the lamination is zero, hence *M* has an essential torus. We strongly suspect this is actually true in all cases. We hope that this method generalizes to the context of (4,1)-surfaces in any 3-manifold. This would establish some kind of finiteness property for (4,1)-surfaces in a 3-manifold, as in the case of incompressible surfaces.

Item Type: | Thesis (Dissertation (Ph.D.)) | ||||
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Subject Keywords: | Mathematics | ||||

Degree Grantor: | California Institute of Technology | ||||

Division: | Physics, Mathematics and Astronomy | ||||

Major Option: | Mathematics | ||||

Thesis Availability: | Public (worldwide access) | ||||

Research Advisor(s): | - Gabai, David
| ||||

Thesis Committee: | - Gabai, David (chair)
- Bonahon, Francis
- Candel, Alberto
- Hersonsky, Saar
- Ramakrishnan, Dinakar
| ||||

Defense Date: | 5 January 1998 | ||||

Other Numbering System: |
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Record Number: | CaltechTHESIS:11202019-173910635 | ||||

Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:11202019-173910635 | ||||

DOI: | 10.7907/yw04-qx32 | ||||

Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||

ID Code: | 13586 | ||||

Collection: | CaltechTHESIS | ||||

Deposited By: | Melissa Ray | ||||

Deposited On: | 21 Nov 2019 17:33 | ||||

Last Modified: | 19 Apr 2021 22:32 |

## Thesis Files

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