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

This article is cited in 10 scientific papers (total in 10 papers)

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.

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

Bibliographic databases:

MSC: 57Rxx
Received: 22.02.2006

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
\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
\jour St. Petersburg Math. J.
\yr 2007
\vol 18
\issue 5
\pages 809--CCCXXXV

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    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    3. Silver D.S., Williams S.G., “Alexander-Lin twisted polynomials”, J. Knot Theory Ramifications, 20:3 (2011), 427–434  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi
    5. Dunfield N.M., Friedl S., Jackson N., “Twisted Alexander Polynomials of Hyperbolic Knots”, Exp. Math., 21:4 (2012), 329–352  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    7. Kohno T., Pajitnov A., “Novikov Homology, Jump Loci and Massey Products”, Cent. Eur. J. Math., 12:9 (2014), 1285–1304  crossref  mathscinet  zmath  isi  scopus
    8. Friedl S., Vidussi S., “the Thurston Norm and Twisted Alexander Polynomials”, J. Reine Angew. Math., 707 (2015), 87–102  crossref  mathscinet  zmath  isi  scopus
    9. Friedl S., Maxim L., “Twisted Novikov homology of complex hypersurface complements”, Math. Nachr., 290:4 (2017), 604–612  crossref  mathscinet  zmath  isi  scopus
    10. Friedl S., “Novikov homology and non-commutative Alexander polynomials”, J. Knot Theory Ramifications, 26:2 (2017), 1740013  crossref  mathscinet  zmath  isi  scopus
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