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Topology of Closed One-Forms
 
Michael Farber Tel Aviv University, Tel Aviv, Israel and University of Durham, Durham, England
Topology of Closed One-Forms
Hardcover ISBN:  978-0-8218-3531-9
Product Code:  SURV/108
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1335-4
Product Code:  SURV/108.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-0-8218-3531-9
eBook: ISBN:  978-1-4704-1335-4
Product Code:  SURV/108.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
Topology of Closed One-Forms
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Topology of Closed One-Forms
Michael Farber Tel Aviv University, Tel Aviv, Israel and University of Durham, Durham, England
Hardcover ISBN:  978-0-8218-3531-9
Product Code:  SURV/108
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1335-4
Product Code:  SURV/108.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-0-8218-3531-9
eBook ISBN:  978-1-4704-1335-4
Product Code:  SURV/108.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
  • Book Details
     
     
    Mathematical Surveys and Monographs
    Volume: 1082004; 246 pp
    MSC: Primary 58; 57;

    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").
    Readership

    Graduate students and research mathematicians interested in geometry and topology.

  • Table of Contents
     
     
    • Chapters
    • 1. The Novikov numbers
    • 2. The Novikov inequalities
    • 3. The universal complex
    • 4. Construction of the universal complex
    • 5. Bott-type inequalities
    • 6. Inequalities with Von Neumann Betti numbers
    • 7. Equivariant theory
    • 8. Exactness of the Novikov inequalities
    • 9. Morse theory of harmonic forms
    • 10. Lusternik-Schnirelman theory, closed 1-forms, and dynamics
    • Appendix A. Manifolds with corners
    • Appendix B. Morse-Bott functions on manifolds with corners
    • Appendix C. Morse-Bott inequalities
    • Appendix D. Relative Morse theory
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1082004; 246 pp
MSC: Primary 58; 57;

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").
Readership

Graduate students and research mathematicians interested in geometry and topology.

  • Chapters
  • 1. The Novikov numbers
  • 2. The Novikov inequalities
  • 3. The universal complex
  • 4. Construction of the universal complex
  • 5. Bott-type inequalities
  • 6. Inequalities with Von Neumann Betti numbers
  • 7. Equivariant theory
  • 8. Exactness of the Novikov inequalities
  • 9. Morse theory of harmonic forms
  • 10. Lusternik-Schnirelman theory, closed 1-forms, and dynamics
  • Appendix A. Manifolds with corners
  • Appendix B. Morse-Bott functions on manifolds with corners
  • Appendix C. Morse-Bott inequalities
  • Appendix D. Relative Morse theory
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.