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- Lecturer
- Andrei Pajitnov

- Place

- Credits
- Lecture1 Exercise0 Experiment0

- Code
- 11081

- Syllabus updated
- 2014/12/26

- Lecture notes updated
- 2014/12/19

- Semester
- Fall Semester

1) Novikov complex for circle-valued Morse functions.

2) Witten-type de Rham framework for Novikov inequalities.

3) Morse-Novikov theory for 3-knots

4) Applications to symplectic topology.

Classical Morse theory establishes a relation between the number

of critical points of a real-valued function on a manifold and

the topology of the manifold.

Circle-valued Morse theory is a branch of the Morse theory.

It originated from a problem in hydrodynamics studied by

S. Novikov in the early 1980s.

Nowadays it is a constantly growing field of geometry with

applications and connections to many geometric problems

such as Arnold's conjectures in symplectic topology,

fibrations of manifolds over the circle, dynamical zeta

functions, and the theory of 3-dimensional knots and links.

縲

Our course will start with recollections on the classical Morse theory

(gradient flows, Morse complex, Morse inequalities).

Then we proceed to the circle-valued Morse theory and construct

the Novikov complex (the generalization of the Morse complex to

the case of circle-valued functions).

This chain complex is generated over the Laurent series ring

by the critical points of the circle-valued function.

E. Witten showed in the beginning of 1980s that the classical Morse

theory can be reformulated in the framework of the de Rham theory.

We will explain his work, and its generalization to the Morse-Novikov

theory, which leads to a relation between the Novikov homology and

the homology with local coefficients.

A natural application of the circle-valued Morse theory is to the topology

of knots. If a knot K in the 3-dimensional sphere S is not fibred, then

any circle-valued function on S-K has critical points. The minimal number

of these critical points is an invariant of K, called the Morse-Novikov

number MN(K).

We will show how to give computable lower bounds for MN(K) with the help

of the Novikov homology, and relate it to the tunnel number of the knot.

In the last part of the course we will explain how the Novikov ring appears

in the Floer's work on the Arnold conjecture concerning the closed orbits

of periodic Hamiltonians, and discuss some recent developments in this direction.

縲

縲

Mentioned in the class.