Geodesic
The Morse--Novikov Theory of Circle-Valued Functions and Noncommutative Localization
Informatics and Automation, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 381-388.

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We use noncommutative localization to construct a chain complex which counts the critical points of a circle-valued Morse function on a manifold, generalizing the Novikov complex. As a consequence we obtain new topological lower bounds on the minimum number of critical points of a circle-valued Morse function within a homotopy class, generalizing the Novikov inequalities. 
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M. Farber; A. Ranicki. The Morse--Novikov Theory of Circle-Valued Functions and Noncommutative Localization. Informatics and Automation, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 381-388. http://geodesic.mathdoc.fr/item/TRSPY_1999_225_a24/

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