Orateur: Mark
Powell (Université du Québec à Montréal & MPI Bonn) :
Titre: Casson towers and
slice knots.
Résumé:
A Casson tower is a 4-manifold which can be constructed by thickening
a 2-complex built from layers of immersed discs in a bigger 4-manifold. A
Casson tower has a height and an attaching circle, which is the boundary of
its base disc. A higher tower, namely a tower with more layers of immersed
discs, is a better approximation to an embedded disc. Casson towers featured
prominently in Freedman's original proof of the 4-dimensional topological
Poincaré conjecture. In particular Freedman showed that an infinite Casson
tower, which is called a Casson handle, contains an embedded disc. The
ability to embed discs was the key to being able to apply surgery and
h-cobordism techniques, originally from high dimensional topology, to
4-manifolds. I will explain what a Casson tower is in more detail and
present embedding results, from work with Jae Choon Cha, on Casson towers of
height 4, 3 and 2. The height 2 result in particular can be used to find a
new family of topologically slice knots. I will explain why we think these
slice knots are interesting.