On the topology and analysis of a closed one form. I (Novikov's theory revisited)

Abstract

We consider systems $(M,\omega ,g)$ with $M$ a closed smooth manifold, $\omega$ a real valued closed one form and $g$ a Riemannian metric, so that $(\omega ,g)$ is a Morse-Smale pair, Definition~2. We introduce a numerical invariant $\rho (\omega ,g)\in [0,\infty ]$ and improve Morse-Novikov theory by showing that the Novikov complex comes from a cochain complex of free modules over a subring ${\Lambda }_{[\omega ],\rho }^{\prime }$ of the Novikov ring ${\Lambda }_{[\omega ]}$ which admits surjective ring homomorphisms $\ev_s:Λ'_{[ω],ρ}\to\C$ for any complex number $s$ whose real part is larger than $\rho$. We extend Witten-Helffer-Sjöstrand results from a pair $(h,g)$ where $h$ is a Morse function to a pair $(\omega ,g)$ where $\omega$ is a Morse one form. As a consequence we show that if $\rho <\infty$ the Novikov complex can be entirely recovered from the spectral geometry of $(M,\omega ,g)$.

36 pages, 2 figures, AMSTeX