Quantum Topology Seminar
Torsten Asselmeyer-Maluga
German Aerospace Center (DLR)
Foliations of 4-manifolds and hyperbolic geometry
Abstract: Foliations of 4-manifolds and hyperbolic geometry
In this talk I will describe a second bridge to noncommutative geometry, the leaf space of a foliation. Exotic 4-manifolds admit non-trivial codimension-one foliations with non-zero Godbillon-Vey invariants. The leaf space of the foliation is a factor III_1 von Neumann algebra. The Godbillon-Vey invariant is the abelian Chern-Simons theory and the coupling with a Spin_C Dirac spinor leads to the Seiberg-Witten theory. From the geometric point of view, the foliations induces a hyperbolic geometry. According to Morgan and Shalen, the space of hyperbolic structures can be compactifified and I interpret the limit as high-curvature regime in gravity leading to a dimensional reduction (4 to 2). Furthermore the limit has connections to grope cobordisms (to describe concordance classes of knots) to understand the IHX relation, for instance.
Thursday February 4, 2021 at 1:00 PM in Zoom