Morse–Novikov cohomology

In the early 1980s, Sergei Petrovich Novikov constructed an analog of Morse theory for smooth closed $1$-forms on a compact smooth manifold $M$. He proposed Morse-type inequalities (later called Novikov inequalities) for the numbers $m_p(\omega)$ of zeros of index $p$ of an arbitrary smooth closed Morse $1$-form $\omega$ on $M$. S.P. Novikov proposed a method based on Witten's approach to Morse theory for finding the so-called torsion-free Novikov inequalities. The central role in this method is played by the cohomology $H^*_{\omega}(M)$ of the complex of differential forms $\Lambda^*(M)$ of a smooth manifold $M$ with deformed differential $d + \omega$, where $\omega$ is a smooth closed $1$-form on the manifold $M$.
Subsequently, such cohomology began to be called Morse–Novikov cohomology in the literature, and they began to be used in a wide variety of problems, and most often their application did not involve studying the critical points of $1$-forms. The talk will be devoted to a review of the development of Morse–Novikov cohomology theory and its applications.