Arnold conjecture over integers

Abstract

For any closed symplectic manifold, we show that the number of 1-periodic orbits of a nondegenerate Hamiltonian thereon is bounded from below by a version of total Betti number over Z of the ambient space taking account of the total Betti number over Q and torsions of all characteristic. The proof is based on constructing a Hamiltonian Floer theory over the Novikov ring with integer coefficients, which generalizes our earlier work for constructing integervalued Gromov-Witten type invariants. In the course of the construction, we build a Hamiltonian Floer flow category with compatible smooth global Kuranishi charts. This generalizes a recent work of Abouzaid-McLean-Smith, which might be of independent interest. Contents 1. Introduction 1 2. Recap of the FOP natural transformation and multiplicativity 11 3. Abstract constructions of chain complexes and maps over the integers 22 4. Proof of the integral Arnold conjecture 52 5. Global Kuranishi charts on Floer moduli spaces 69 6. Smoothing 118 7. Constructions for PSS, SSP, and the homotopy 146 Appendix A. Product of canonical Whitney stratifications 155 Appendix B. Relative stable equivariant smoothing 160 Appendix C. A proof sketch of Proposition 5.64 163 References 166

Abstract interlevel persistence for Morse-Novikov and Floer theory

“…13 See [Oh15, Definition 19.6.1] for the definition of semipositivity; this condition should not be essential if one adapts virtual techniques as in [BX22], [Par16]. However, we will use in Proposition 7.7 some facts related to the PSS map that, to the best of my knowledge, are not available in the literature in the non-semipositive case.…”