Circle valued Morse theory and Novikov homology. (English) Zbl 1068.57031

Ranicki, Andrew

Farrell, F. Thomas (ed.) et al., Topology of high-dimensional manifolds. Proceedings of the school on high-dimensional manifold topology, Abdus Salam ICTP, Trieste, Italy, May 21–June 8, 2001. Number 1 and 2. Trieste: The Abdus Salam International Centre for Theoretical Physics (ISBN 92-95003-12-8/pbk). ICTP Lect. Notes 9, 539-569 (2002).

This nice paper contains the lectures about circle valued Morse theory and Novikov homology given by the author at the Summer School on High-Dimensional Manifold Topology (Trieste, 21 May-8 June, 2001). The paper is divided into five chapters. In Chapter 1 the author reviews classical real valued Morse theory. Chapters 2-4 introduce the circle valued Morse theory and the universal coefficient versions of the Novikov complex and Novikov homology, which involve the universal cover of the manifold. The last chapter formulates an algebraic chain complex model in the universal coefficient version for the relationship between the Novikov complex defined by a circle valued Morse function and the Morse-Smale complex defined by the associated real valued Morse function on a fundamental domain of the infinite cyclic cover of the manifold. A useful bibliography consisting in 36 suggestive references for the topic is given.
For the entire collection see [Zbl 0996.00038].

Reviewer: Dorin Andrica (Cluj-Napoca)

MSC:

57R70	Critical points and critical submanifolds in differential topology
55N35	Other homology theories in algebraic topology
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

Morse function; gradient-like vector field; Morse-Smale complex; Novikov complex; Novikov homology; Morse-Novikov inequalities; algebraic fundamental domain