Arnold conjecture over integers

TL;DR

This paper proves a lower bound on the number of 1-periodic orbits for Hamiltonians on symplectic manifolds using integer-based Floer theory, extending previous work with new algebraic and geometric tools.

Contribution

It introduces a Hamiltonian Floer theory over the Novikov ring with integer coefficients, generalizing earlier Gromov-Witten invariants and constructing a Floer flow category with Kuranishi charts.

Findings

Bound on 1-periodic orbits related to Betti numbers over Z and Q

Construction of Floer theory over integers with Novikov ring

Development of a Floer flow category with smooth Kuranishi charts

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 integer-valued 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.