Geometry & Topology

Event Information Morse broken trajectories and hyperbolic volume
16:10 on Monday December 14, 2015
17:00 on Monday December 14, 2015
BA2179, Bahen Center, 40 St. George St.
Hannah Alpert

MIT

A large family of theorems all state that if a space is topologically complex, then the functions on that space must express that complexity, for instance by having many singularities. For the theorem in this talk, our preferred measure of topological complexity is the hyperbolic volume of a closed manifold admitting a hyperbolic metric (or more generally, the Gromov simplicial volume of any space). A Morse function on a manifold with large hyperbolic volume may still not have many critical points, but we show that there must be many flow lines connecting those few critical points. Specifically, given a closed n-dimensional manifold and a Morse-Smale function, the number of n-part broken trajectories is at least the Gromov simplicial volume. To prove this we adapt lemmas of Gromov that bound the simplicial volume of a stratified space in terms of the complexity of the stratification. A notable corollary: every Morse function on a closed hyperbolic manifold must have a critical point of every index.