Products of locally conformal symplectic manifolds

Geometry Topology Seminar
Monday, November 13, 2023 - 3:00pm for 1 hour (actually 50 minutes)
University of Georgia (Boyd 303)
Kevin Sackel – UMass Amherst
Austin Christian

Locally conformal symplectic (LCS) geometry is a variant of symplectic geometry in which the symplectic form is locally only defined up to positive scale. For example, for the symplectization R x Y of a contact manifold Y, translation in the R direction are symplectomorphisms up to scale, and hence the quotient (R/Z) x Y is naturally an LCS manifold. The importation of symplectic techniques into LCS geometry is somewhat subtle because of this ambiguity of scale. In this talk, we define a notion of product for LCS manifolds, in which the underlying manifold of an LCS product is not simply the smooth product of the underlying manifolds, but which nonetheless appears to fill the same role in LCS geometry as the standard symplectic product does in standard symplectic geometry. As a proof of concept, with input from an LCS result of Chantraine and Murphy, we use the LCS product to prove that C^0 small Hamiltonian isotopies have a lower bound on the number of fixed points given by the rank Morse-Novikov homology. This is a natural generalization of the classical symplectic proof of the analogous result by Laudenbach and Sikorav which uses the graph of a Hamiltonian diffeomorphism in the product manifold. These results are joint work in progress with Baptiste Chantraine.