Geodesic
The Morse--Novikov Theory of Circle-Valued Functions and Noncommutative Localization
Trudy Matematicheskogo Instituta imeni V.A. Steklova, 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. 
@article{TM_1999_225_a24,
     author = {M. Farber and A. Ranicki},
     title = {The {Morse--Novikov} {Theory} of {Circle-Valued} {Functions} and {Noncommutative} {Localization}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     publisher = {mathdoc},
     volume = {225},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_1999_225_a24/}
}
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M. Farber; A. Ranicki. The Morse--Novikov Theory of Circle-Valued Functions and Noncommutative Localization. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 381-388. http://geodesic.mathdoc.fr/item/TM_1999_225_a24/