On the Novikov complex for rational Morse forms
It is well known that, given a Morse function \(f\) on a smooth manifold \(M\) one can construct a chain complex \(C_*(f)\). The number of free generators of \(C_*(f)\) in dimension \(p\) is exactly the Morse number \(m_p (f)\) of critical points of \(f\) of index \(p\). \textit{S. P. Novikov} in [Russ. Math. Surv. 37, No. 5, 1-56 (1982); translation from Usp. Mat. Nauk 37, No. 5(227), 3-49 (1982; Zbl 0571.58011)] expanded the Morse theory to the case of Morse forms, that is closed 1-forms which are locally the derivatives of Morse functions. The existence of the complex enables to obtain Morse type inequalities for the Morse numbers \(m_p (\omega)\) of zeros of \(\omega\) of index \(p\) in homotopy invariant terms. The aim of the paper under review is to give a complete and detailed proof of sharpness of arising inequalities. More precisely, let \(\omega\) be a closed 1-form with nondegenerate zeros on a closed manifold \(M\) such that de Rham cohomology class \([\omega]\) is nonzero and is up to the multiplication by a constant, an element of \(H^1(M, \mathbb{Z})\). Let \(\widehat p:\widehat M \to M\) be a regular connected covering with the structure group \(G\), such that \(\widehat p^* ([\omega]) = 0\). To this data the author associates a free based complex \(C_*\) over the Novikov completion of \(\mathbb{Z} G\). For every \(p\) the number of free generators of \(C_p\) is equal to the Morse number \(m_p (\omega)\). The simple homotopy type of \(C_*\) is equal to the simple homotopy type of the Novikov-completed simplicial chain complex of \(\widehat M\).
scientific article; zbMATH DE number 824219
scientific article