Integral Floer Homology Theory

(joint work with Shaoyun Bai) Using a new version of transversality condition (the FOP transversality) on orbifolds, one can construct Hamiltonian Floer theory over integers for all compact symplectic manifolds. In this talk I will first describe the formal structures in this theory, including the formalism of flow categories, bimodules, homotopies, their lifts to more refined categories such as derived orbifolds. The FOP perturbation method allows us to deduce from such formal structures the usual objects in Floer theory: the Floer chain complex, continuation map, PSS map, pair-of-pants product, Steenrod-type operations, etc. I will mention two dynamical applications for any compact symplectic manifold: 1) the existence of a non-contractible periodic orbit of a Hamiltonian implies infinitely many such orbits; 2) the existence of Hamiltonian pseudorotation implies geometric uniruledness, i.e., there exists a rational curve through each point.