Burghelea, Dan; Haller, Stefan 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]. Reviewer: Corina Mohorianu (Iaşi) Cited in 9 Documents MSC: 57R70 Critical points and critical submanifolds in differential topology 58J10 Differential complexes 58J50 Spectral problems; spectral geometry; scattering theory on manifolds PDFBibTeX XMLCite \textit{D. Burghelea} and \textit{S. Haller}, Monogr. Enseign. Math. 38, 133--175 (2001; Zbl 1017.57013) Full Text: arXiv