Micro-conference on the Floer fundamendal group

 April 3, 2021.

The recording of the talks and the slides can be found here .




9:00 - 10:00  (Montreal time)

François Charette (Marianapolis College):  Morse Novikov homology and the Arnol'd conjecture for symplectic isotopies

On a closed symplectic manifold M, generic Hamiltonian isotopies have at least as many 1 periodic orbits as M has Betti numbers, by the Arnol'd conjecture.  It is natural to try and extend the result to (non exact) symplectic isotopies.  However,  these do not necessarily have any 1 periodic orbit, e.g. an irrational rotation of the torus.   Nevertheless, Lê-Ono have defined a Floer homology for such symplectic isotopies and shown that it is isomorphic to the Morse-Novikov homology of M associated to the Calabi invariant. In the first part of this micro \pi_1 conference, I will introduce Morse Novikov homology of closed one forms, by using circle valued Morse theory.  Time permitting, I will give a few basic notions of Floer homology for symplectic isotopies, laying the ground for Barraud's talk that will follow.

10:30 - 11:30 (Montreal time)

Jean-François Barraud (Université de Toulouse): Floer-Novikov fundamental group for symplectic isotopies

Abstract: Floer theory explains how the homology of the ambient manifold forces some symplectic phenomena, like fixed points for Hamiltonian isotopies. As explained by H.V. Le and K. Ono (or M. Damian and A. Gadbled in the Lagrangian case), in the case of symplectic but non hamiltonian isotopies, similar results hold where the usual homology is replaced by the Novikov homology associated to the Calabi invarant of the isotopy. I will explain how this picture extends to the fundamental group: I will quickly review how to describe the fundamental group in Morse theory and how to cook up a Novikov version of  it that keeps track of a given degree 1 cohomology class. Then I will discuss how to recover these groups from Floer theoretic objects, at least in the good cases.





Organizer: Octav Cornea (octav.cornea@gmail.com)