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Morse-Novikov theory, Heegaard splittings, and closed orbits of gradient flows


Authors: H. Goda, H. Matsuda and A. Pajitnov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 3.
Journal: St. Petersburg Math. J. 26 (2015), 441-461
MSC (2010): Primary 57M27
Published electronically: March 20, 2015
MathSciNet review: 3289179
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Abstract: The work of Donaldson and Mark made the structure of the Seiberg-Witten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle-valued Morse map on a 3-manifold. In the paper, these invariants are studied by using the Morse-Novikov theory and Heegaard splitting for sutured manifolds, and detailed computations are made for knot complements.


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Additional Information

H. Goda
Affiliation: Department of Mathematics, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei, Tokyo 184-8588, Japan
Email: goda@cc.tuat.ac.jp

H. Matsuda
Affiliation: Department of Mathematical Sciences, Yamagata University, Yamagata 990-8560, Japan
Email: matsuda@sci.kj.yamagata-u.ac.jp

A. Pajitnov
Affiliation: Laboratoire de Mathématiques, Jean-Leray UMR 6629, Université de Nantes, Faculté des Sciences, 2, rue de la Houssinière, 44072, Nantes, Cedex, France
Email: andrei.pajitnov@univ-nantes.fr

DOI: https://doi.org/10.1090/S1061-0022-2015-01345-9
Keywords: Oriented knot, sutured manifold, Morse map, Novikov complex, half-transversal gradients, Lefschetz zeta function
Received by editor(s): March 2, 2013
Published electronically: March 20, 2015
Additional Notes: The first and second authors were partially supported by Grant-in-Aid for Scientific Research (No. 21540071 and No. 20740041), Ministry of Education, Science, Sports and Technology, Japan
Article copyright: © Copyright 2015 American Mathematical Society