The $E$-cohomological Conley Index, Cup-Lengths and the Arnold Conjecture on $T^{2n}$

TL;DR

This paper presents a shorter proof of the Arnold conjecture on tori by utilizing the $E$-cohomological Conley index with a new module structure and cup-length approach to estimate critical points.

Contribution

It introduces a natural module structure for the $E$-cohomological Conley index and applies it to provide a novel, concise proof of the Arnold conjecture on tori.

Findings

Established the $E$-cohomological Conley index as a module.

Derived a new cup-length lower bound for critical points.

Provided a shorter proof of the Arnold conjecture on $T^{2n}$.

Abstract

We give a new proof of the strong Arnold conjecture for $1$-periodic solutions of Hamiltonian systems on tori, that was first shown by C. Conley and E. Zehnder in 1983. Our proof uses other methods and is shorter than the previous one. We first show that the $E$-cohomological Conley index, that was introduced by the first author recently, has a natural module structure. This yields a new cup-length and a lower bound for the number of critical points of functionals. Then an existence result for the $E$-cohomological Conley index, which applies to the setting of the Arnold conjecture, paves the way to a new proof of it on tori.