CaltechTHESIS
  A Caltech Library Service

On the Geometric Simple-Connectivity of 4-Manifolds

Citation

Gadgil, Siddhartha (1999) On the Geometric Simple-Connectivity of 4-Manifolds. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/js0q-8h61. https://resolver.caltech.edu/CaltechTHESIS:11212019-150249902

Abstract

This work concerns the question of when contractible 4-manifolds need 1-handles. In high-dimensions, eliminating handles whenever permitted by homotopy type has been a very fruitful approach. The question addressed concerns the applicability of these methods in low dimensions.

Specifically, we study when the interiors of compact, contractible 4-manifolds have a handle-decomposition without 1-handles. An argument of Casson shows that the compact manifolds themselves 'usually' need 1-handles. This argument depends essentially on finiteness of the handle-decomposition.

We show that any handle-decomposition without 1-handles must be of a particularly nice form, which involves surgery along surfaces representing homology. We show that we have 'Casson finiteness', i.e., Casson’s argument can be used to show that such a handle-decomposition cannot exist, whenever there are embedded, disjoint surfaces satisfying a certain property. We then show that there are immersed surfaces satisfying this property. Finally, we show that the obstruction to cutting and pasting the surfaces to get embedded ones is non-trivial.

As a corollary to the methods, we give an example of an open manifold with an infinite handle-decomposition without 1-handles that is not the interior of a compact manifold, and thus has no finite handle-decomposition.

A relative version of this question is also considered. In this case, Donaldson's theorem leads to obstructions to the existence of finite handle decomposition without 1-handles.

Item Type:Thesis (Dissertation (Ph.D.))
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
Defense Date:6 April 1999
Other Numbering System:
Other Numbering System NameOther Numbering System ID
UMI9932839
Funders:
Funding AgencyGrant Number
Alfred P. Sloan FoundationUNSPECIFIED
Record Number:CaltechTHESIS:11212019-150249902
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:11212019-150249902
DOI:10.7907/js0q-8h61
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13590
Collection:CaltechTHESIS
Deposited By: Melissa Ray
Deposited On:21 Nov 2019 23:26
Last Modified:19 Apr 2021 22:39

Thesis Files

[img]
Preview
PDF - Final Version
See Usage Policy.

1MB

Repository Staff Only: item control page