The Arnold conjecture via Symplectic Field Theory polyfolds

The Arnold conjecture via Symplectic Field Theory polyfolds -Ben Filippenko

Ben Filippenko
University of California, Berkeley
April 1, 2019

I will explain a polyfold proof, joint with Katrin Wehrheim, of the Arnold conjecture: the number of 1-periodic orbits of a nondegenerate 1-periodic Hamiltonian on a closed symplectic manifold is at least the sum of the Betti numbers. Our proof is a polyfold construction of the PSS morphisms between Morse and Floer theory. To construct these maps, we first identify the PSS moduli spaces as fiber products of Morse moduli spaces with Symplectic Field Theory (SFT) moduli spaces. We then use SFT polyfolds to perturb these fiber products into general position. This perturbation required the development of general fiber product constructions in polyfold theory which are of interest for other applications.