On the Geometric Simple-Connectivity of 4-Manifolds

Author: Gadgil, Siddhartha

Year: 1999

Degree: Dissertation (Ph.D.)

Advisor: Gabai, David

Committee Members: Gabai, David; Bonahon, Francis; Candel, Alberto; Hersonsky, Saar

Option: Mathematics

DOI: 10.7907/js0q-8h61

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.

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