Topology of Closed One-Forms (Mathematical Surveys & Monographs) by Michael Farber (PDF)

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This monograph is an introduction to the fascinating field of the topology, geometry and dynamics of closed one-forms. The subject was initiated by S. P. Novikov in 1981 as a study of Morse type zeros of closed one-forms. The first two chapters of the book, written in textbook style, give a detailed exposition of Novikov theory, which plays a fundamental role in geometry and topology. Subsequent chapters of the book present a variety of topics where closed one-forms play a central role. The most significant results are the following: The solution of the problem of exactness of the Novikov inequalities for manifolds with the infinite cyclic fundamental group. The solution of a problem raised by E. Calabi about intrinsically harmonic closed one-forms and their Morse numbers. The construction of a universal chain complex which bridges the topology of the underlying manifold with information about zeros of closed one-forms. This complex implies many interesting inequalities including Bott-type inequalities, equivariant inequalities, and inequalities involving von Neumann Betti numbers. The construction of a novel Lusternik-Schnirelman-type theory for dynamical systems. Closed one-forms appear in dynamics through the concept of a Lyapunov one-form of a flow. As is shown in the book, homotopy theory may be used to predict the existence of homoclinic orbits and homoclinic cycles in dynamical systems (“”focusing effect””).

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