| Circle-valued Morse Theory Contributor(s): Pajitnov, Andrei V. (Author) |
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ISBN: 3110158078 ISBN-13: 9783110158076 Publisher: de Gruyter OUR PRICE: $285.00 Product Type: Hardcover - Other Formats Published: December 2006 Annotation: In 1927 M. Morse discovered that the number of critical points of a smooth function on a manifold is closely related to the topology of the manifold. This became a starting point of the Morse theory which is now one of the basic parts of differential topology. Reformulated in modern terms the geometric essence of Morse theory is as follows. For a smooth function f on a closed manifold having only non-degenerate critical points (a Morse function) there is a chain complex M (the Morse complex) freely generated by the set of all critical points of f, such that the homology of M is isomorphic to the homology of the manifold. The boundary operators in this complex are related to the geometry of the gradient flow of the function. It is natural to consider also circle-valued Morse functions, that is, smooth functions with values in S1 having only non-degenerate critical points. The study of such functions was initiated by S. P. Novikov in the early 1980s in relation to a problem in hydrodynamics. The formulation of the circle-valued Morse theory is a new branch of topology. At present the Morse--Novikov theory is a large and actively developing domain of differential topology, with applications and connections to many geometrical problems like the Arnol'd conjecture in the theory of Lagrangian intersections, fibrations of manifolds over the circle, dynamical zeta functions, and the theory of knots and links in S3. The aim of the present book is to give a systematic treatment of the geometric foundations of the subject, and of some recent research results. The book is accessible for first year graduate students specializing in geometry and topology. |
| Additional Information |
| BISAC Categories: - Mathematics | Geometry - Differential - Mathematics | Mathematical Analysis |
| Dewey: 514.72 |
| Series: de Gruyter Studies in Mathematics |
| Physical Information: 1.16" H x 6.89" W x 9.6" (1.99 lbs) 463 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: In 1927 M. Morse discovered that the number of critical points of a smooth function on a manifold is closely related to the topology of the manifold. This became a starting point of the Morse theory which is now one of the basic parts of differential topology. It is a large and actively developing domain of differential topology, with applications and connections to many geometrical problems. The aim of the present book is to give a systematic treatment of the geometric foundations of a subfield of that topic, the circle-valued Morse functions, a subfield of Morse theory. |