Circle-valued Morse Theory
Reformulated in modern terms, the geometric essence of Morse theory is as follows. For a smooth function f on a closed manifold having only non-degenerate critical points (a Morse function), there is a chain complex MCC (Morse chain complex) freely generated by the set of critical points of f, such that the homology of MCC is isomorphic to the homology of the underlying manifold and the boundary operator in this complex is related to the geometry of the gradient flow of f. Motivated by a problem in hydrodynamics, S. P. Novikov initiated a study of circle-valued Morse functions in the early 80s. The aim of the book is to give a systematic treatment of the geometric foundations of Morse-Novikov theory. Various applications of this approach to problems in differential topology include the Arnold Conjecture in the theory of Lagrangian intersections, fibrations of manifolds over the circle, dynamical zeta functions and the theory of knots and links in the three sphere.