Mathematical Surveys and Monographs 2004; 246 pp; hardcover Volume: 108 ISBN10: 0821835319 ISBN13: 9780821835319 List Price: US$76 Member Price: US$60.80 Order Code: SURV/108
 This monograph is an introduction to the fascinating field of the topology, geometry and dynamics of closed oneforms. The subject was initiated by S. P. Novikov in 1981 as a study of Morse type zeros of closed oneforms. 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 oneforms 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 oneforms 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 oneforms. This complex implies many interesting inequalities including Botttype inequalities, equivariant inequalities, and inequalities involving von Neumann Betti numbers.
 The construction of a novel LusternikSchnirelmantype theory for dynamical systems. Closed oneforms appear in dynamics through the concept of a Lyapunov oneform 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").
Readership Graduate students and research mathematicians interested in geometry and topology. Table of Contents  The Novikov numbers
 The Novikov inequalities
 The universal complex
 Construction of the universal complex
 Botttype inequalities
 Inequalities with von Neumann Betti numbers
 Equivariant theory
 Exactness of the Novikov inequalities
 Morse theory of harmonic forms
 LusternikSchnirelman theory, closed 1forms, and dynamics
 Appendix A. Manifolds with corners
 Appendix B. MorseBott functions on manifolds with corners
 Appendix C. MorseBott inequalities
 Appendix D. Relative Morse theory
 Bibliography
 Index
