Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued functions and functionals: An analogue of the Morse Theory) began to study Morse theory of closed $1$-forms, a special case of which is circle-valued Morse Theory (the subject of a very nice book by Pajitnov). The key new idea was to use the Novikov ring, a certain ring of formal power series, as a tool to count gradient flow lines of an induced Morse function on a covering space of $M$.
Novikov's ideas have been quite influential, but I have not been able to find a good introductory text on general Morse-Novikov theory just by googling keywords. My interests are in classical differential topology, in low dimensions. The reason that I want such a reference is that I work quite a bit with finite covering spaces (not necessarily abelian covers), and I'm curious as to whether or not Morse-Novikov theory gives me a better intuitive handle on what is happening than just plain old garden-variety Morse theory.
Reference Request: Can anyone recommend/ Does there exist a good introductory text on Morse-Novikov Theory?
Best Answer
S. P. Novikov, An analogue of Morse theory for many-valued functions. Certain properties of Poisson brackets. Appendix I in Dubrovin, Novikov, and Fomenko, Modern Geometry-Methods and Applications Part III. Introduction to Homology Theory. This short paper was written for the student working through their textbook treatment of differential geometry.