Interactions of 3- and 4-Dimensional Topology

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

Topology is the study of abstract spaces with properties that are unchanged by deformations such as bending, stretching, and twisting, but not breaking, and low-dimensional topology involves spaces (called manifolds) that locally look like 2-, 3-, or 4-dimensional Euclidean space. Low-dimensional topology has a wide range of real-world applications, including deep connections to theoretical physics and quantum computing. In addition, topology has applications to data science; researchers understand how to interpret a large data set as a topological space in order to gain insights by studying the topological shape of that data. While our world is 3-dimensional in nature, the evolution of a 3-dimensional object over a period of time gives rise to a 4-dimensional space, in which time constitutes a fourth degree of freedom. Topology in dimension three has seen an explosion of activity over the last several decades, reaching a peak with Grigory Perelman's proof of the Poincaré conjecture in 2003, the only resolved problem of the seven Millennium Problems posed by the Clay Mathematics Institute. In contrast, the topology of 4-dimensional manifolds has become an increasingly active area of research, with many fundamental problems still open. This research project involves adapting ideas from dimension three to discover novel approaches to these 4-dimensional problems. The award provides funds to support research by graduate students. The award will also support the Great Plains Alliance, a program initiated by the PI to connect graduate students with speaking opportunities at nearby institutions. Two foundational open problems in 4-manifold topology are the smooth 4-dimensional Poincaré Conjecture, which asserts that every closed 4-manifold homotopy equivalent to the standard 4-sphere is diffeomorphic to the standard 4-sphere, and the slice-ribbon conjecture, which posits that any knot bounding an embedded disk in the 4-ball also bounds an immersed ribbon disk in the 3-sphere. This project describes a varied set of problems interweaving ideas from knot theory, 3-manifolds, and smooth 4-manifold topology. The PI’s prior work includes two major collaborative contributions in this area, demonstrating that a large family of homotopy 4-spheres is diffeomorphic to the standard smooth 4-sphere, and developing far-reaching connections between 3-dimensional structures and 4-manifold trisections. The current work involves four main problems. First, the PI will characterize a family of knots satisfying an intermediate condition between being slice and being ribbon. Second, the PI will pursue a new approach to the Powell conjecture, concerning the generation of the group of diffeomorphisms of the 3-sphere leaving a Heegaard surface invariant. Third, the PI will use bridge trisections to address a Kirby problem related to representing second homology classes of 4-manifolds with smoothly embedded 2-spheres, and finally, the PI will adapt ideas from Heegaard splittings of 3-manifolds to trisections of 4-manifolds in order to better understand handle decompositions of exotic 4-manifolds. The underlying theme of this work is an approach that simultaneously integrates 3- and 4-dimensional techniques in order to obtain new insights into long-standing open problems. 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.