Trisections of non-orientable 4-manifolds

Abstract

Broadly, this thesis is concerned with trying to understand 4-manifolds through 3-dimensional techniques. From the point of view of smooth manifolds, dimension four is quite unique; one striking illustration of this is the fact that R^n admits either one (if n is not equal to four) or uncountably many (if is equal to four) smooth structures. There are many remaining questions about the differences between topological and smooth categories in dimension four: for instance, the last remaining Poincaré conjecture asks whether S^4 admits a unique smooth structure. Nonetheless, one might attempt to use tools from lower dimensions to gain some insight.

One highly useful tool in dimension three is the notion of a Heegaard splitting: a symmetric decomposition of a closed 3-manifold into two handlebodies that meet along an embedded closed surface. Originally introduced by Heegaard in 1898, they connect 3-manifolds to fundamental objects like mapping class groups groups and the curve complex. Recent techniques like Heegaard Floer homology theories have shown that they are also an effective computational tool that can be used to distinguish 3-manifolds.

In analogy with Heegaard splittings, Gay and Kirby recently introduced the idea of a trisection of an orientable closed 4-manifold: a decomposition into three 4-dimensional handlebodies with controlled intersection data. Because a trisection is largely determined by lower-dimensional information, one would hope to use 3-dimensional techniques to understand 4-dimensional phenomena. Trisections have already been used to reprove fundamental results in gauge theory, and define new invariants for 4-manifolds. In this thesis, we complete the theory of trisections for non-orientable 4-manifolds.

Chapter 1 gives the necessary preliminaries for the rest of the thesis. Chapter 2 is a self-contained introduction to trisections that summarizes the current state of the literature and contains many motivating examples.

Chapter 3 is concerned with developing the 3- and 4-dimensional results necessary to carefully extend the theory of trisections to the non-orientable setting. In particular, we prove an analogue of a theorem of Laudenbach-Poénaru which does not seem to appear in the literature. We also give a non-orientable version of Waldhausen’s theorem on Heegaard splittings of #S^2×S^1 which may be of independent interest.

In Chapter 4, we extend the theory of trisections to non-orientable 4-manifolds. By adapting the orientable case and results from Chapter 3, we give proofs of existence and stable uniqueness, along with many examples. We also cover non-orientable relative trisections (for 4-manifolds with boundary) and bridge trisections (for embedded surfaces in 4-manifolds).