RUI: New Approaches to Understanding the Four-Sphere

Western Washington University, Bellingham WA

Investigators

Abstract

We inhabit a three-dimensional space. This means that, locally, one can move in three independent directions: left/right, forward/backward, and up/down. However, the global nature of our three-dimensional space is not known. For example, it is not known whether our universe extends infinitely far in every direction or if it curls back on itself. To complicate things, it becomes more natural to view the space we inhabit as a four-dimensional space by including time. The goal of low-dimensional topology is to study and classify three-dimensional and four-dimensional spaces. One important aspect of this program is to understand the topology (or "shape") of the four-dimensional sphere, which is a four-dimensional analog the spherical surface of a solid ball. Amazingly, despite over one hundred years of study, very little is known about the topology of the four-dimensional sphere. In contrast, the analogous objects in dimensions greater than or less than four are much better understood. The goal of this research project is to apply novel techniques and approaches better understand four-dimensional sphere. This work will help to inform and advance the general study of four-dimensional spaces, such as the spacetime expanse that we inhabit. The award provides funds to support research training for undergraduate students. Perhaps the most important open problem in low-dimensional topology, the Smooth Poincare Conjecture, ask whether any smooth four-manifold with the homotopy-type of the four-sphere is equivalent to the four-sphere. This problem has been open for more than 100 years, yet has seen almost no progress. Rather than approach this problem directly, this research program aims to study the smooth topology from four distinct, but related, angles. One angle is to study four-manifolds with the homotopy-type of the four-sphere that admit simple handle-decompositions. Recently, the PI and Zupan gave new results in this setting, offering the first general progress towards the Poincare Conjecture in 30 years. The first aim of this research is to build on this work to obtain results that apply in even broader settings. In another direction, the introduction of theory of trisections by Gay and Kirby in 2016, and the subsequent development and expansion, especially by the PI and Zupan, has ushered in a new era in four-manifold topology, with applications emerging to many interesting aspects of low-dimensional topology. This research project aims to advance and refine these theories, broaden their applicability, and bring to fruition many applications and connections. In particular, the PI will investigate what sort of trisections the four-sphere can admit. In the process, connections will be established with important open problems in low-dimensional topology, such as the Andrews-Curtis, Generalized Property R, and Slice-Ribbon Conjectures. The project will include an investigation of trisections of genus three, where the additivity of trisection genus will be explored in search for exotic copies of familiar four-manifolds. Finally, the PI will employ the theory of bridge trisections to study knotted surfaces in the four-sphere. This theory was introduced by the PI and Zupan and offers a powerful new way to study knot theory in dimension four. In particular, the PI will work to give connections between the braid group and potentially exotic knotted spheres. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.