Transversal Homotopy of Spheres and the Tangle Hypothesis
Jon Woolf (University of Liverpool)
Monday 21st February, 2011 16:00-17:00 204
Transversal homotopy theory assigns invariants to a stratified space X by considering smooth maps from disks into X which are transversal to all strata. As an example, the third transversal homotopy category of the 2-sphere with a marked point is equivalent (essentially by the Pontrjagin-Thom construction) to the category of framed tangles. Baez-Dolan's Tangle Hypothesis is that `the n-category of codimension k, framed n-tangles is equivalent to the free k-tuply monoidal n-category with duals on one object'. For n=1 and k=2 this is a theorem, due to Shum: usually phrased as `the category of framed tangles is the free ribbon category'. The aim of this talk is to explain one interpretation of all these terms using a recent, very geometric, definition of n-category due to Morrison and Walker, and to sketch out the relationship between the transversal homotopy theory of spheres with marked points and the Tangle Hypothesis.