Topology of closed one-forms / Michael Farber.

Format:

Book

Author/Creator:

Farber, Michael, 1951- author.

Series:

Mathematical surveys and monographs ; no. 108.

Mathematical surveys and monographs, 0076-5376 ; volume 108

Language:

English

Subjects (All):

Critical point theory (Mathematical analysis).

Differential topology.

Foliations (Mathematics).

Physical Description:

1 online resource (262 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2004]

Language Note:

English

Summary:

Viewed locally, a closed one-form on a manifold is a smooth function up to an additive constant. The global structure of a closed one-form is mainly determined by its de Rham cohomology class. In this book, Michael Farber studies fascinating geometrical, topological, and dynamical properties of closed one-forms. In particular, he reveals the relations between their global and local features. In 1981, S. P. Novikov initiated a generalization of Morse theory in which, instead of critical points of smooth functions, one deals with closed one-forms and their zeros. The first two chapters of the book, written in textbook style, give a detailed exposition of Novikov theory, which now plays a fundamental role in geometry and topology. In the following chapters the author describes the universal chain complex that lives over a localization (in the sense of P. M. Cohn) of the group ring and relates the topology of the underlying manifold with information about zeros of closed one-forms. Using this complex, many different variations and generalizations of the Novikov inequalities are obtained, including Bott-type inequalities for closed one-forms, equivariant inequalities, and inequalities involving von Neumann Betti numbers. Another significant result in the book is a solution of the problem about exactness of the Novikov inequalities for manifolds with the infinite cyclic fundamental group. One of the chapters deals with the problem raised by E. Calabi about intrinsically harmonic closed one-forms and their Morse numbers. Presented are the solution of this problem and a detailed study of topological properties of singular foliations of closed one-forms. The last chapter suggests a new Lusternik-Schnirelman-type theory for closed one-forms. The dynamics of the gradient-like flows play a crucial role in this theory. As is shown here, homotopy theory may be used to predict the existence of homoclinic orbits and homoclinic cycles in dynamical systems (``focusing effect''). The book is suitable for graduate students and researchers interested in geometry and topology.

Contents:

""Contents""; ""Preface""; ""Chapter 1. The Novikov Numbers""; ""1.1. Homological algebra of Morse inequalities""; ""1.2. The Novikov ring . . . ""; ""1.3. The rational subring R(Γ)""; ""1.4. Homology of local coefficient systems""; ""1.5. The Novikov numbers""; ""1.6. Further properties of the Novikov numbers""; ""1.7. Novikov numbers and Betti numbers of flat line bundles""; ""Chapter 2. The Novikov Inequalities""; ""2.1. Closed 1-forms""; ""2.2. Geometry of Novikov theory""; ""2.3. The Novikov inequalities""; ""Chapter 3. The Universal Complex""; ""3.1. The Main Theorem""

""3.2. Line bundles and algebraic integers""""3.3. Generic flat vector bundles""; ""3.4. Examples""; ""Chapter 4. Construction of the Universal Complex""; ""4.1. Chain collapse""; ""4.2. Proof of Theorem 3.1 in the rank 1 case""; ""4.3. Proof of Theorem 3.1 in the general case""; ""4.4. Refined universal complex and deformation complex""; ""Chapter 5. Bott-type Inequalities""; ""5.1. Topology of the set of zeros""; ""5.2. Proofs of Theorems 5.1 and 5.5""; ""Chapter 6. Inequalities with Von Neumann Betti Numbers""; ""Chapter 7. Equivariant Theory""; ""7.1. Basic 1-forms""

""7.2. Equivariant Novikov inequalities""""7.3. Application: Fixed points of a symplectic circle action""; ""7.4. Signature via Novikov numbers""; ""Chapter 8. Exactness of the Novikov Inequalities""; ""8.1. Exactness Theorem""; ""8.2. Finiteness theorem for codimension two knots""; ""8.3. Surgery on codimension one submanifolds""; ""8.4. Algebra of minimal lattices""; ""8.5. Proof of The Exactness Theorem""; ""Chapter 9. Morse Theory of Harmonic Forms""; ""9.1. Topology of singular foliations of closed 1-forms""; ""9.2. Intrinsically harmonic 1-forms""

""9.3. Examples of singular foliations""""9.4. Proof of Calabi's Theorem""; ""9.5. Morse numbers of harmonic 1-forms""; ""Chapter 10. Lusternik-Schnirelman Theory, Closed 1-Forms,and Dynamics""; ""10.1. Colliding the critical points""; ""10.2. Closed 1-forms on topological spaces""; ""10.3. Category of a space with respect to a cohomology class""; ""10.4. Estimate of the number of zeros""; ""10.5. Gradient-convex neighborhoods""; ""10.6. Movable homology classes""; ""10.7. Cohomological lower bound for cat(X, ξ)""; ""10.8. Deformations and their spectral sequences""

""10.9. Families of flat bundles and higher Massey products""""10.10. Estimate for . . .""; ""10.11. Flows, Lyapunov 1-forms and asymptotic cycles""; ""Appendix A. Manifolds with Corners""; ""Appendix B. Morse-Bott Functions on Manifolds with Corners""; ""Appendix C. Morse-Bott Inequalities""; ""Appendix D. Relative Morse Theory""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""L""; ""M""; ""N""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""Z""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 239-244) and index.

Description based on print version record.

ISBN:

1-4704-1335-3