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

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