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Topology is concerned with the study of manifolds, which are spaces which locally look like Euclidean space (think a line, or a plane, or the three dimensional space around you). However manifolds are globally more complicated than Euclidean space; for example a circle looks like a line when you zoom in, thus is a 1-manifold, but unlike a line a circle has the property that if an ant living in the space went for a long walk in one direction it might end up back where it started. Other examples of interesting manifolds include the glaze on a donut (which looks like a plane when you zoom in) or the universe people build when they play the computer game Portal (which looks like normal three space for the player, but has these ‘extra’ tunnels).
I am particularly interested in three and four dimensional manifolds. Because these spaces are globally pretty complicated we often study them by cutting them up into simpler pieces. Conversely we often construct complicated manifolds by taking simple pieces and gluing them together in some strange way. I study the relationships between distinct decompositions of the same manifold.
One might hope that when a manifold can be constructed from distinct sets of pieces there must be certain relationships between the pieces used. For example an old conjecture of Akbulut and Kirby asserted that if a pair of knots have homeomorphic 0-surgeries (this is some three manifold constructed using the knot) than the knots must be smoothly concordant. This is false, a counterexample was given by Yasui in 2015. However, a corrected conjecture was immediately proposed, namely that if a pair of knots have diffeomorphic 0-traces (this is some four manifold constructed using the knot) than the knots must be smoothly concordant. In 2017 A.N. Miller and I disproved this corrected conjecture. Our paper appeared in the Journal of Topology, and a preprint is available here.
In later work, I showed that in fact there exist pairs of knots with diffeomorphic 0-traces and distinct slice genera. This is an improvement on my work with Miller, and has a significant consequence. The slice genus of a knot gives an upper bound on the complexity of an important submanifold of a knot trace. There is a Kirby problem (1.41) conjecturing that the bound coming from the slice genus is not sharp. My examples prove this, since they produce distinct upper bounds. That preprint is available here.
An advantage of studying multiple constructions of the same manifold is that various properties of the manifold may be apparent under one construction and not another. Seminal work of Eliashberg showed that a four manifold admits a Stein structure if and only if it admits a handle decomposition satisfying certain requirements. It is not hard to show that being Stein does not imply that every handle decomposition satisfies these requirements, but it was an open question in the field whether all ‘simple’ handle decompositions of Stein knot traces meet the requirements. By studying distinct simple handle decompositions of knot traces T. Mark, F. Vafaee and I showed that there are simple handle decompositions of Stein knot traces that do not meet the requirements. Our preprint is available here.
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