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Hardcover ISBN:  9780821835319 
Product Code:  SURV/108 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470413354 
Product Code:  SURV/108.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821835319 
eBook ISBN:  9781470413354 
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 DetailsMathematical Surveys and MonographsVolume: 108; 2004; 246 ppMSC: Primary 58; 57
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").
ReadershipGraduate 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. Botttype inequalities

6. Inequalities with Von Neumann Betti numbers

7. Equivariant theory

8. Exactness of the Novikov inequalities

9. Morse theory of harmonic forms

10. 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


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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").
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. Botttype inequalities

6. Inequalities with Von Neumann Betti numbers

7. Equivariant theory

8. Exactness of the Novikov inequalities

9. Morse theory of harmonic forms

10. 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