Novikov homology, twisted Alexander polynomials, and Thurston cones
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- by A. V. Pajitnov
- St. Petersburg Math. J. 18 (2007), 809-835
- DOI: https://doi.org/10.1090/S1061-0022-07-00975-2
- Published electronically: August 10, 2007
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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 $\xi :H\to \mathbf R$ 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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Bibliographic Information
- A. V. Pajitnov
- Affiliation: Laboratoire Mathématiques Jean Leray, UMR 6629, Université de Nantes, Faculté des Sciences, 2, Rue de la Houssinière, 44072, Nantes, Cedex, France
- Email: pajitnov@math.univ-nantes.fr
- Received by editor(s): February 22, 2006
- Published electronically: August 10, 2007
- © Copyright 2007 American Mathematical Society
- Journal: St. Petersburg Math. J. 18 (2007), 809-835
- MSC (2000): Primary 57Rxx
- DOI: https://doi.org/10.1090/S1061-0022-07-00975-2
- MathSciNet review: 2301045