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 MorseSmale pair, Definition~2. We introduce a numerical invariant $\rho(\omega,g)\in[0,\infty]$ and improve MorseNovikov theory by showing that the Novikov complex comes from a cochain complex of free modules over a subring $\Lambda'_{[\omega],\rho}$ of the Novikov ring $\Lambda_{[\omega]}$ which admits surjective ring homomorphisms $\ev_s:\Lambda'_{[\omega],\rho}\to\C$ for any complex number $s$ whose real part is larger than $\rho$. We extend WittenHelfferSjöstrand 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 $(M,\omega,g)$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2001
 arXiv:
 arXiv:math/0101043
 Bibcode:
 2001math......1043B
 Keywords:

 Mathematics  Differential Geometry;
 58G26
 EPrint:
 36 pages, 2 figures, AMSTeX