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On the topology and analysis of a closed one form. I. (Novikov’s theory revisited). (English) Zbl 1017.57013

Ghys, Étienne (ed.) et al., Essays on geometry and related topics. Mémoires dédiés à André Haefliger. Vol. 1. Genève: L’Enseignement Mathématique. Monogr. Enseign. Math. 38, 133-175 (2001).
A system \((M,\omega,g)\), consisting of a closed connected smooth \(n\)-dimensional manifold \(M\), a closed one form \(\omega\) and a Riemannian metric \(g\) is considered. If \((\omega,g)\) satisfies the Morse-Smale condition, the Novikov incidence (integer) numbers \(I_q(x,y,\widehat{\alpha})\) can be algebraically organized to produce a cochain complex \((WC^*,\partial^*)\) of free modules over the Novikov ring \(\Lambda_{[\omega]}\).
In the first part of this paper the authors introduce a numerical invariant \(\rho(\omega,g)\in [0,\infty]\) and, using the theory of Dirichlet series and extensions of Witten-Helffer-Sjöstrand results, improve Morse-Novikov theory for \((M,\omega,g)\), provided \(\rho(\omega,g)<\infty\). They show that the Novikov complex of \((M,\omega,g)\) is an extension of a complex \((C^*,\partial_s^*)\) of free modules over a ring \(\Lambda_{[\omega],\rho}'\subset \Lambda_{[\omega]}\) of holomorphic functions in \(\{s\in\mathbb{C}\mid \text{Re}(s)>\rho\}\). The \(q\)-component \(C^q\) is the \(\mathbb{C}\)-vector space generated by the critical points of \(\omega\) of index \(q\). The boundaries \(\partial^q_s\) are defined by the Dirichlet series obtained from the numbers \(I_{q+1}(x,y,\widehat{\alpha})\).
The authors also construct a smooth one-parameter family of cochain complexes of finite dimensional \(\mathbb{R}\)-vector spaces and a smooth family of isomorphisms from \((\Omega^*_{\ell,sm}(M),d_t^*)\) to \((\text{Maps}(Cr_*(\omega),\mathbb{R}),\partial_t^*)\). Then \((\Omega^*_{t,s}(M),d^*_t)\) will carry all information provided by the Novikov complex \((N\mathbb{C}^*,\partial^*)\).
If \((w,g)\) satisfies the Morse-Smale conditions, \(\rho(w,g)<\infty\) and the orientations \(o\), the authors reformulate and extend results of Helffer and Sjöstrand for \(\omega\) exact.
The family \((\Omega^*_{t,sm}(M),d^*_t)\) can be viewed as an analytic substitute of the Novikov complex and the Novikov incidence numbers \(I_q(x,y,\widehat{\alpha})\) can be entirely recovered from the spectral geometry of \((M,\omega,g)\).
New results with their proofs are included in this paper and other pleasant applications are announced to be worked out in a second part of this work.
For the entire collection see [Zbl 0988.00114].

MSC:

57R70 Critical points and critical submanifolds in differential topology
58J10 Differential complexes
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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