RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2006, Volume 18, Issue 5, Pages 173–209 (Mi aa93)

Research Papers

Novikov homology, twisted Alexander polynomials, and Thurston cones

A. V. Pajitnov

Laboratoire Mathématiques Jean Leray, Université de Nantes, Faculté des Sciences, Nantes

Abstract: Let $M$ be a connected CW complex, and let $G$ denote the fundamental group of $M$. Let $\pi$ be an epimorphism of $G$ onto a free finitely generated Abelian group $H$, let $\xi\colon H\to\mathbf R$ be a homomorphism, and let $\rho$ be an antihomomorphism of $G$ to the group $\operatorname{GL}(V)$ of automorphisms of a free finitely generated $R$-module $V$ (where $R$ is a commutative factorial ring).
To these data, we associate the twisted Novikov homology of $M$, which is a module over the Novikov completion of the ring $\Lambda=R[H]$. The twisted Novikov homology provides the lower bounds for the number of zeros of any Morse form whose cohomology class equals $\xi\circ\pi$. This generalizes a result by H. Goda and the author.
In the case when $M$ is a compact connected 3-manifold with zero Euler characteristic, we obtain a criterion for the vanishing of the twisted Novikov homology of $M$ in terms of the corresponding twisted Alexander polynomial of the group $G$.
We discuss the relationship of the twisted Novikov homology with the Thurston norm on the 1-cohomology of $M$.
The electronic preprint of this work (2004) is available from the ArXiv.

Full text: PDF file (357 kB)
References: PDF file   HTML file

English version:
St. Petersburg Mathematical Journal, 2007, 18:5, 809–CCCXXXV

Bibliographic databases:

MSC: 57Rxx
Language:

Citation: A. V. Pajitnov, “Novikov homology, twisted Alexander polynomials, and Thurston cones”, Algebra i Analiz, 18:5 (2006), 173–209; St. Petersburg Math. J., 18:5 (2007), 809–CCCXXXV

Citation in format AMSBIB
\Bibitem{Paj06} \by A.~V.~Pajitnov \paper Novikov homology, twisted Alexander polynomials, and Thurston cones \jour Algebra i Analiz \yr 2006 \vol 18 \issue 5 \pages 173--209 \mathnet{http://mi.mathnet.ru/aa93} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2301045} \zmath{https://zbmath.org/?q=an:1137.57017} \elib{https://elibrary.ru/item.asp?id=9295936} \transl \jour St. Petersburg Math. J. \yr 2007 \vol 18 \issue 5 \pages 809--CCCXXXV \crossref{https://doi.org/10.1090/S1061-0022-07-00975-2} 

• http://mi.mathnet.ru/eng/aa93
• http://mi.mathnet.ru/eng/aa/v18/i5/p173

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Morifuji T., “Twisted Alexander polynomials of twist knots for nonabelian representations”, Bull. Sci. Math., 132:5 (2008), 439–453
2. Papadima S., Suciu A.I., “Bieri-Neumann-Strebel-Renz invariants and homology jumping loci”, Proc. Lond. Math. Soc. (3), 100:3 (2010), 795–834
3. Silver D.S., Williams S.G., “Alexander-Lin twisted polynomials”, J. Knot Theory Ramifications, 20:3 (2011), 427–434
4. Friedl S., Vidussi S., “Twisted Alexander Polynomials and Fibered 3-Manifolds”, Low-Dimensional and Symplectic Topology, Proceedings of Symposia in Pure Mathematics, 82, ed. Usher M., Amer Mathematical Soc, 2011, 111–130
5. Dunfield N.M., Friedl S., Jackson N., “Twisted Alexander Polynomials of Hyperbolic Knots”, Exp. Math., 21:4 (2012), 329–352
6. Friedl S., Vidussi S., “A Vanishing Theorem for Twisted Alexander Polynomials with Applications to Symplectic 4-Manifolds”, J. Eur. Math. Soc., 15:6 (2013), 2027–2041
7. Kohno T., Pajitnov A., “Novikov Homology, Jump Loci and Massey Products”, Cent. Eur. J. Math., 12:9 (2014), 1285–1304
8. Friedl S., Vidussi S., “the Thurston Norm and Twisted Alexander Polynomials”, J. Reine Angew. Math., 707 (2015), 87–102
9. Friedl S., Maxim L., “Twisted Novikov homology of complex hypersurface complements”, Math. Nachr., 290:4 (2017), 604–612
10. Friedl S., “Novikov homology and non-commutative Alexander polynomials”, J. Knot Theory Ramifications, 26:2 (2017), 1740013
•  Number of views: This page: 284 Full text: 80 References: 41 First page: 3