Integral Arnol'd Conjecture

TL;DR

This paper adapts advanced geometric and topological methods to Hamiltonian Floer theory, enabling the construction of Floer complexes and proving the Arnold conjecture over integers through a new framework of equivariant manifolds.

Contribution

It introduces the concept of $ angle k angle$-manifolds and develops smoothing and perturbation theories to construct Floer complexes, proving the Arnold conjecture integrally.

Findings

Constructed global Kuranishi charts for Floer trajectories.

Proved the Arnold conjecture over the integers.

Outlined a bifurcation analysis framework for Floer invariants.

Abstract

We explain how to adapt the methods of Abouzaid-McLean-Smith to the setting of Hamiltonian Floer theory. We develop a language around equivariant ``$\langle k \rangle$-manifolds'', which are a type of manifold-with-corners that suffices to capture the combinatorics of Floer-theoretic constructions. We describe some geometry which allows us to straightforwardly adapt Lashofs's stable equivariant smoothing theory and Bau-Xu's theory of FOP-perturbations to $\langle k \rangle$-manifolds. This allows us to compatibly smooth global Kuranishi charts on all Hamiltonian Floer trajectories at once, in order to extract a Floer complex and prove the Arnol'd conjecture over the integers. We also make first steps towards a further development of the theory, outlining the analog of bifurcation analysis in this setting, which can give short independence proofs of the independence of Floer-theoretic invariants of all choices involved in their construction.