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Let M be a weakly monotone symplectic manifold, and H be a time-dependentHamiltonian; we assume that the periodic orbits of the correspondingtime-dependent Hamiltonian vector field are non-degenerate. We construct arefined version of the Floer chain complex associated to these data and anyregular covering of M, and derive from it new lower bounds for the number ofperiodic orbits. We prove in particular that if the fundamental group of M isfinite and solvable or simple, then the number of periodic orbits is not lessthan the minimal number of generators of the fundamental group. For a generalclosed symplectic manifold with infinite fundamental group, we show theexistence of 1-periodic orbit of Conley-Zehnder index 1-n for anynon-degenerate 1-periodic Hamiltonian system.

On the fixed points of a Hamiltonian diffeomorphism in presence of fundamental group