Preliminary version (written by D.Burghelea) of the introduction to the paper LAPLACE TRANSFORM, TOPOLOGY AND SPECTRAL GEOMETRY (OF A CLOSED ONE FORM)

We consider a system (M, g, ω,X) where (M, g) is a closed Riemannian manifold, ω a closed one form whose all zeros are nondegenerate and X an (−ω, g) gradient like vector field. For t ∈ R+ large enough, the Witten complex (Ω∗, dt = d ∗ + tω∧) decomposes canonically as a direct sum of two one parameter family of cochain complexes (Ωsm(t), dt ) and (Ωla(t), d ∗ t ). We show that under a mild hypotheses, generically satisfied, the inverse Laplace transform of (Ωsm(t), d∗(t)) determines completely the Novikov’s instantons counting (organized as a Dirichlet series), Theorem 3, and the inverse Laplace transform of the analytic torsion of (Ωla(t), d ∗(t)) properly corrected, determines the counting function of closed trajectories of X.