We live in a 4-dimensional universe, counting time as a dimension, and yet surprisingly we understand very little about the geometry of 4-dimensional spaces. Topology studies truly fundamental geometric issues about space that are unchanged even by smoothly bending and deforming the structure of space. As an example from three dimensions, a loop of string with a square knot tied into it is topologically distinct from a loop of string with a granny knot tied into it, in the sense that no amount of smooth deformation of the string or the space around it can turn one loop into the other; studying this kind of knottedness is the quintessential three-dimensional topological problem. Knottedness also happens in four dimensions, in numerous different ways, and is also at the core of smooth four-dimensional topology, and thus really at the core of understanding the shape and anatomy of exactly the kind of space that we spend our lives inhabiting. The research supported by this award will probe these problems by looking at four-dimensional topology from multiple different dimensions, ranging from a careful study of projections of four-dimensional spaces and their subspace onto two dimensional spaces to a careful study of motions of four-dimensional spaces inside five- and six-dimensional spaces. This research is also closely related to an outreach component, in which the principal investigator will work with design professionals and members of the local community to create exciting visualizations of important ideas in topology, to broaden the public's awareness of and interest in these fascinating topics. In addition the project provides research training opportunities for graduate students.
Two of the biggest remaining problems in topology are the smooth 4-dimensional Poincare and Schoenflies conjectures. (Poincare: Every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. Schoenflies: Every smoothly embedded 3-sphere in 4-space bounds a smoothly embedded 4-ball.) A related and foundational problem is whether the smooth mapping class group of the 4-sphere is trivial. Existing techniques such as gauge theory say nothing in the context of these problems; the basic problem is that there seems to be "nothing to hold on to" when working with spheres or even homotopy spheres; low-dimensional topology needs new ideas. In collaboration first with Kirby and later with a larger group, the PI, over the course of several earlier NSF awards, developed the study of Morse 2-functions, generalizing Morse theory, and then developed the theory of trisections of 4-manifolds as a striking 4-dimensional analog to the theory of Heegaard splittings. This seems to be exactly what is needed to inject new life into 4-manifold topology and to allow for extensive cross-fertilization between the worlds of 3-manifolds and 4-manifolds. The past five years (2014-19), in particular, have seen an explosion of interest by other researchers in trisections, and a significant broadening of the community involved in this work. The PI also has a strong background in contact and symplectic topology; the proposed project with Licata builds on this work, using many of the same tools used in the work on trisections. Here are two long-shot motivational examples: It is conceivable that understanding the space of all trisections on the 4-sphere will allow us to show that certain self-diffeomorphisms are not isotopic to the identity, and also perhaps distinguish the 4-sphere from other homotopy 4-spheres. Likewise it is possible that a deeper understanding of contact topology in relation to 4-manifold topology, perhaps via trisections and open book decompositions, will allow us to see every embedded 3-sphere in 4-space as being of contact type, allowing us to use symplectic tools to settle the Schoenflies conjecture.
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.
