Pure Maths Colloquium: The disparity between smooth and topologically slice knots
2 October 2017 16:00 in CM221
Understanding the subtle difference between the topological and smooth categories is a central subject in 4-manifold topology. The local difference in the categories can be studied through knot theory. A knot in the 3-sphere is called smoothly slice (respectively, topologically slice) if it bounds a smoothly embedded (respectively, topologically embedded, locally flat) disc in the 4-ball.
After Casson used the celebrated results of Freedman and Donaldson on 4-dimensional manifolds to find a topologically slice knot that is not smoothly slice, distinguishing topologically slice knots and smoothly slice knots has been extensively studied.
In 2012, in order to try to introduce some structure into the problem, Cochran, Harvey and Horn introduced a remarkable framework to study the set (actually a group) of topologically slice knots modulo smoothly slice knots, defining a geometrically inspired filtration, called the bipolar filtration. However a crucial question left unsettled was the non-triviality of the graded quotients. In joint work with Jae Choon Cha, we showed that the graded quotient of the bipolar filtration in fact has infinite rank, at each stage greater than one. In this talk, I will give a friendly, short introduction to knot theory in 3 and 4 dimensions, focussing on the differences between the smooth and topological categories, and I will briefly survey the recent progress on the bipolar filtration.
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