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Non-commutative Reidemeister torsion and Morse-Novikov theory


Author: Takahiro Kitayama
Journal: Proc. Amer. Math. Soc. 138 (2010), 3345-3360
MSC (2010): Primary 57Q10; Secondary 57R70
DOI: https://doi.org/10.1090/S0002-9939-10-10418-3
Published electronically: April 30, 2010
MathSciNet review: 2653964
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Abstract: Given a circle-valued Morse function of a closed oriented manifold, we prove that Reidemeister torsion over a non-commutative formal Laurent polynomial ring equals the product of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of the Novikov complex over the ring. This paper gives a generalization of the result of Hutchings and Lee on abelian coefficients to the case of skew fields. As a consequence we obtain a Morse theoretical and dynamical description of the higher-order Reidemeister torsion.


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  • 1. T. Cochran, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004) 347-398. MR 2077670 (2005k:57023)
  • 2. S. Friedl, Reidemeister torsion, the Thurston norm and Harvey's invariants, Pacific J. Math. 230 (2007) 271-296. MR 2309160 (2008c:57021)
  • 3. R. Geoghegan and A. Nicas, Trace and torsion in the theory of flows, Topology 33 (1994) 683-719. MR 1293306 (95h:55002)
  • 4. H. Goda and A. V. Pajitnov, Dynamics of gradient flows in the half-transversal Morse theory, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009) 6-10. MR 2488751 (2009k:57048)
  • 5. H. Goda and T. Sakasai, Homology cylinders in knot theory, arXiv:0807.4034.
  • 6. S. Harvey, Higher-order polynomial invariants of $ 3$-manifolds giving lower bounds for the Thurston norm, Topology 44 (2005) 895-945. MR 2153977 (2006g:57019)
  • 7. M. Hutchings, Reidemeister torsion in generalized Morse theory, Forum Math. 14 (2002) 209-244. MR 1880912 (2003f:57047)
  • 8. M. Hutchings and Y. J. Lee, Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of three manifolds, Topology 38 (1999) 861-888. MR 1679802 (2000i:57055)
  • 9. M. Hutchings and Y. J. Lee, Circle-valued Morse theory and Reidemeister torsion, Geom. Topol. 3 (1999) 369-396. MR 1716272 (2000h:57063)
  • 10. B. J. Jiang and S. C. Wang, Twisted topological invariants associated with representations, Topics in knot theory (Erzurum, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 399, Kluwer Acad. Publ., Dordrecht (1993) 211-227. MR 1257911
  • 11. X. S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engle. Ser.) 17 (2001) 361-380. MR 1852950 (2003f:57018)
  • 12. J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358-426. MR 0196736 (33:4922)
  • 13. J. Milnor, Infinite Cyclic Coverings, in Conference on the Topology of Manifolds, ed. Hocking, Prindle, Weber & Schmidt, Boston, MA (1968). MR 0242163 (39:3497)
  • 14. S. P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Soviet Math. Dokl. 24 (1981) 222-226 (1982). MR 630459 (83a:58025)
  • 15. D. Passman, The Algebraic Structure of Group Rings, Wiley, New York (1977). MR 470211 (81d:16001)
  • 16. A. V. Pajitnov, Closed orbits of gradient flows and logarithms of non-abelian Witt vectors, K-Theory 21 (2000) 301-324. MR 1828180 (2002e:57048)
  • 17. A. V. Pajitnov, Circle-Valued Morse Theory, de Gruyter Studies in Mathematics, 32, Walter de Gruyter & Co., Berlin (2006) MR 2319639 (2008c:58008)
  • 18. A. V. Pazhitnov, On the Novikov complex for rational Morse forms, Annales de la Faculté de Science de Toulouse 4 (1995) 297-338. MR 1344724 (97f:57033)
  • 19. A. V. Pazhitnov, The simple homotopy type of the Novikov complex, and the Lefschetz $ \zeta$-function of the gradient flow, Uspekhi Mat. Nauk 54 (1999) 117-170; translation in Russian Math. Surveys 54 (1999) 119-169. MR 1706835 (2001m:57041)
  • 20. A. V. Pazhitnov, On closed orbits of gradient flows of circle-valued mappings, Algebra i Analiz 14 (2002) 186-240; translation in St. Petersburg Math. J. 14 (2003) 499-534. MR 1921994 (2003g:57052)
  • 21. D. Schütz, Gradient flows of closed $ 1$-forms and their closed orbits, Forum Math. 14 (2002) 509-537. MR 1900172 (2003h:57044)
  • 22. D. Schütz, One-parameter fixed point theory and gradient flows of closed $ 1$-forms, K-Theory 25 (2002) 59-97. MR 1899700 (2003g:57055)
  • 23. E. Spanier, Algebraic Topology, Springer-Verlag, corrected reprint (1989). MR 666554 (83i:55001)
  • 24. V. Turaev, Introduction to Combinatorial Torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2001). MR 1809561 (2001m:57042)
  • 25. V. Turaev, Torsions of $ 3$-dimensional manifolds, Progress in Mathematics, 208, Birkhäuser Verlag, Basel (2002). MR 1958479 (2003m:57028)
  • 26. M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994) 241-256. MR 1273784 (95g:57021)

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Additional Information

Takahiro Kitayama
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
Email: kitayama@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-10-10418-3
Keywords: Reidemeister torsion, Morse-Novikov complex, derived series
Received by editor(s): September 2, 2009
Received by editor(s) in revised form: December 28, 2009
Published electronically: April 30, 2010
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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