# ANNALES DE L'INSTITUT FOURIER

Fibered cohomology classes in dimension three, twisted Alexander polynomials and Novikov homology
Annales de l'Institut Fourier, Online first, 28 p.

We prove that for “most” closed $3$-dimensional manifolds $M$, the existence of a closed non singular one-form in a given cohomology class $u\in {H}^{1}\left(M,ℝ\right)=\mathrm{Hom}\left({\pi }_{1}\left(M\right),ℝ\right)$ is equivalent to the fact that every twisted Alexander polynomial ${\Delta }^{H}\left(M,u\right)\in ℤ\left[G/keru\right]$ associated to a normal subgroup with finite index $H<{\pi }_{1}\left(M\right)$ has a unitary $u$-minimal term.

Nous prouvons que pour « la plupart » des variétés fermées de dimension trois, l’existence d’une forme fermée non singulière dans une classe de cohomologie donnée $u\in {H}^{1}\left(M,ℝ\right)=\mathrm{Hom}\left({\pi }_{1}\left(M\right),ℝ\right)$ équivaut au fait que tout polynôme d’Alexander tordu ${\Delta }^{H}\left(M,u\right)\in ℤ\left[G/keru\right]$ associé à un sous-groupe distingué d’indice fini $H<{\pi }_{1}\left(M\right)$ a un terme $u$-minimal unitaire.

Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3531
Classification: 57K30,  57K14,  57M05,  57M10,  20C07,  20E26,  20F19,  20F65,  20J05
Keywords: Three-manifolds, fibrations, Alexander polynomials, Novikov homology
Sikorav, Jean-Claude 1

1 Unité de Mathématiques Pures et Appliquées UMR CNRS 5669 École normale supérieure de Lyon (France)
@unpublished{AIF_0__0_0_A100_0,
author = {Sikorav, Jean-Claude},
title = {Fibered cohomology classes in dimension three, twisted {Alexander} polynomials and {Novikov} homology},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
year = {2022},
doi = {10.5802/aif.3531},
language = {en},
note = {Online first},
}
TY  - UNPB
AU  - Sikorav, Jean-Claude
TI  - Fibered cohomology classes in dimension three, twisted Alexander polynomials and Novikov homology
JO  - Annales de l'Institut Fourier
PY  - 2022
DA  - 2022///
PB  - Association des Annales de l’institut Fourier
N1  - Online first
UR  - https://doi.org/10.5802/aif.3531
DO  - 10.5802/aif.3531
LA  - en
ID  - AIF_0__0_0_A100_0
ER  - 
%0 Unpublished Work
%A Sikorav, Jean-Claude
%T Fibered cohomology classes in dimension three, twisted Alexander polynomials and Novikov homology
%J Annales de l'Institut Fourier
%D 2022
%I Association des Annales de l’institut Fourier
%Z Online first
%U https://doi.org/10.5802/aif.3531
%R 10.5802/aif.3531
%G en
%F AIF_0__0_0_A100_0
Sikorav, Jean-Claude. Fibered cohomology classes in dimension three, twisted Alexander polynomials and Novikov homology. Annales de l'Institut Fourier, Online first, 28 p.

[1] Agol, Ian The virtual Haken conjecture, with an appendix by Agol, I., D. Groves, J. Mannings, Doc. Math., Volume 18 (2013), pp. 1045-1087 | Zbl

[2] Bieri, Robert; Neumann, Walter D.; Strebel, Ralph A geometric invariant of discrete groups, Invent. Math., Volume 90 (1987), pp. 451-477 | DOI | MR | Zbl

[3] Eisenbud, David Commutative Algebra. With a View Towards Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer, 1995 | Zbl

[4] Eizenbud, A.; Lichtman, Alexander I. On embedding of group rings of residually torsion-free nilpotent groups into division rings, Trans. Am. Math. Soc., Volume 299 (1987), pp. 373-386 | DOI | Zbl

[5] Friedl, Stefan; Vidussi, Stefano, The Mathematics of Knots: Theory and Application (Contributions in Mathematical and Computational Sciences), Volume 1 (2008), pp. 45-94

[6] Friedl, Stefan; Vidussi, Stefano Twisted Alexander polynomials and symplectic structures, Am. J. Math., Volume 130 (2008) no. 2, pp. 455-484 | DOI | MR | Zbl

[7] Friedl, Stefan; Vidussi, Stefano A vanishing theorem for twisted Alexander polynomials with applications to symplectic $4$-manifolds, J. Eur. Math. Soc., Volume 15 (2013), pp. 2027-2041 | MR | Zbl

[8] Goldie, Alfred W. The structure of prime rings under ascending chain conditions, Proc. Lond. Math. Soc., Volume 8 (1958), pp. 589-608 | DOI | MR | Zbl

[9] Hall, Philip Finiteness conditions for soluble groups, Proc. Lond. Math. Soc., Volume 4 (1954), pp. 419-436 | DOI | MR | Zbl

[10] Hall, Philip On the finiteness of certain soluble groups, Proc. Lond. Math. Soc., Volume 9 (1959), pp. 595-622 | DOI | MR | Zbl

[11] Kargapolov, Mikhail I.; Merzliakov, Ju. I. Fundamentals of the theory of groups. Translated from the second Russian edition by Robert G. Burns, Graduate Texts in Mathematics, 62, Springer, 1979

[12] Kielak, Dawid Residually finite rationally solvable groups and virtual fibring, J. Am. Math. Soc., Volume 33 (2020) no. 2, pp. 451-486 | DOI | MR | Zbl

[13] Koberda, Thomas Residual properties of fibered and hyperbolic manifolds, Topology Appl., Volume 160 (2013) no. 7, pp. 857-886 | MR | Zbl

[14] Kochloukova, Dessislava H. Some Novikov rings that are von Neumann finite, Comment. Math. Helv., Volume 81 (2006) no. 4, pp. 931-943 | DOI | Zbl

[15] Lam, Tsit-Yuen Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer, 1998

[16] Milnor, John W., Conference on the Topology of Manifolds. E. Lansing, 1967 (The Prindle, Weber and Schmidt Complementary Series in Mathematics), Volume 13 (1968), pp. 115-133 | Zbl

[17] Passman, Donald S. The algebraic structure of group rings, John Wiley & Sons, 1977 | MR

[18] Robinson, Derek J. S. A Course in the Theory of Groups. Second Edition, Graduate Texts in Mathematics, 80, Springer, 1995

[19] Sikorav, Jean-Claude Homologie de Novikov associée à une classe de cohomologie de degré un (quatrième partie de la thèse), Ph. D. Thesis, Université Paris-Sud (1987)

[20] Sikorav, Jean-Claude Novikov homology, 2017

[21] Stallings, John, Topology of 3-manifolds and related topics (1962), pp. 95-100 | MR | Zbl

[22] Thurston, William P. A norm for the homology of $3$-manifolds, Two papers: Genera of the arborescent links and A norm for the homology of 3-manifolds (Memoirs of the American Mathematical Society), Volume 339, American Mathematical Society, 1986, p. i-vi and 99–130 | Zbl

[23] Tischler, David On fibering certain foliated manifolds over ${S}^{1}$, Topology, Volume 9 (1970), pp. 153-154 | DOI | MR | Zbl

Cited by Sources: