PhD thesis defence

Monday, August 9, 2021 10:00 am - 10:00 am EDT (GMT -04:00)

Patrick Naylor, Department of Pure Mathematics, University of Waterloo

"Trisections of non-orientable 4—manifolds"

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 4) or uncountably many (if n=4) smooth structures. There are many remaining fundamental questions about four dimensional topology, but one might hope to use lower dimensional tools to gain some insight. 

One highly useful such tool is the notion of a Heegaard splitting: a decomposition of a 3—manifold into two equal pieces. In analogy with this decomposition, Gay and Kirby recently defined the notion of a trisection of a closed oriented 4—manifold: it is a similar decomposition, but into three equal pieces. Trisections provide a novel diagrammatic perspective on 4—manifolds, and have already been used to define new invariants and reprove fundamental results in gauge theory. 

In this talk, I will discuss my thesis (based on a joint project with Maggie Miller), which extends these decompositions to non-orientable 4—manifolds. In particular, we prove a non-orientable analogue of a theorem of Laudenbach-Poénaru that does not seem to appear in the literature, and is critical for descriptions of closed non-orientable 4—manifolds. We also give a non-orientable analogue of a theorem of Waldhausen on Heegaard splitting off #S^2xS1. The talk will begin with a brief introduction to the theory of trisections, and then highlight the differences in the non-orientable setting. 

Please contact Nancy Maloney ( if you are interested in virtually attending Patrick's talk.