Three-dimensional topology and some four-dimensional contexts

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Scharlemann's recent research has centered on the theory of classical knots and of 3-manifolds, in particular on the use of Heegaard splittings and of related notions (eg bridge positionings, tunnel number) from classical knot theory. The focus is typically on the behavior of surfaces contained in the 3-manifolds (a classical approach) but in a more sophisticated way. Added to old-fashioned combinatorial arguments on surface intersections are ideas from graph theory; added to the classic study of hierarchies on 3-manifolds is Gabai's notion of sutured manifold decomposition, in which parameterizing surfaces and estimates of the Thurston norm help control and understand the topology of the hierarchy; and, added to the classic tool of Morse theory, is the minimax principle of thin position, in which handles of a given index are added not all at once, but as slowly as possible. Recently Scharlemann has gotten interested in how these and similar new tools can also be used towards resolving questions in the topology of 4-manifolds. For example, his recent proof of the genus three Schoenflies Conjecture began with an effort to prove the full Schoenflies Conjecture with, among other ideas, a 4-dimensional application of thin position. The proof of the genus three case (which includes ideas on the general problem and its connection to Property R) also integrates Heegaard theory into this important 4-dimensional problem through the natural use of Heegaard unions. These may be only the first steps of a useful application of 3-manifold ideas to those intriguing topological questions which are sometimes called (3 + 1)-dimensional because they ask how 3- and 4-dimensional manifolds are interrelated. A focus of Scharlemann's interest for many years has been the topology of 3-manifolds. To explain: one of the most basic observations about the world around us, apparent almost from our birth, is that it is 3-dimensional. So it is of interest to understand objects with precisely this property: anyone living in one would see their world as 3-dimensional. Such objects are called ``3-manifolds", and the broad goal of this research proposal is to increase our understanding of them. Of particular (but not sole) interest is what our emerging understanding of 3-dimensional manifolds can tell us about some old and important questions concerning 4-dimensional manifolds. Such 4-manifolds also connect to our natural experience, when we incorporate time as well as space into our thinking. The particular emphasis in this proposal is on questions that sit on the edge between 3- and 4-dimensional manifold theory. Both of these dimensions are interesting in part because they model the universe in which we live.