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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Novikov homology, twisted Alexander polynomials, and Thurston cones

Author: A. V. Pajitnov
Original publication: Algebra i Analiz, tom 18 (2006), nomer 5.
Journal: St. Petersburg Math. J. 18 (2007), 809-835
MSC (2000): Primary 57Rxx
Published electronically: August 10, 2007
MathSciNet review: 2301045
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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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Additional 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

Received by editor(s): February 22, 2006
Published electronically: August 10, 2007
Article copyright: © Copyright 2007 American Mathematical Society