Les invariants θ p des 3-variétés périodiques
[The θ p invariants of periodic 3-manifolds]
Annales de l'Institut Fourier, Volume 51 (2001) no. 4, pp. 1135-1150.

Let r2 be an integer. A 3-manifold M is said to be r-periodic if and only if the group G=/r acts smoothly on M with a circle as the set of fixed points. The aim of this paper is to study the invariants θ p (M) in the case where M is an r-periodic 3-homology sphere. We use the regularity of the Kauffman bracket of periodic links introduced by Murasugi, to find a relationship between the invariant of M and the invariant of the quotient 3-homology sphere M ¯. As an application it is shown that the Poincaré space is not the regular r-fold branched covering of S 3 , if r is a prime congruent to ±1 modulo 5.

Soit r un entier >1. Une 3-variété M est dite r-périodique si et seulement si le groupe cyclique G=/r agit semi-librement sur M avec un cercle comme l’ensemble des points fixes. Dans cet article, nous utilisons les invariants quantiques θ p pour établir des conditions nécessaires pour qu’une 3-variété soit périodique.

DOI: 10.5802/aif.1848
Classification: 57M27
Mot clés : 3-variété périodique, entrelacs périodique, sphère d'homologie, invariants quantiques
Keywords: periodic 3-manifold, periodic link, homology sphere, quantum invariants

Chbili, Nafaa 1

1 Faculté des Sciences de Monastir, Département de Mathématiques, Boulevard de l'Environnement, Monastir 5000 (Tunisie)
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Chbili, Nafaa. Les invariants $\theta _p$ des 3-variétés périodiques. Annales de l'Institut Fourier, Volume 51 (2001) no. 4, pp. 1135-1150. doi : 10.5802/aif.1848. https://aif.centre-mersenne.org/articles/10.5802/aif.1848/

[MR] G. Masbaum; J. Roberts A simple proof of integrality of quantum invariants at prime roots of unity, Math. Proc. Cambridge. Phil. Soc., Volume 121 (1997), pp. 443-454 | DOI | MR | Zbl

[1] J.K. Bartoszynska; J. Przytycki 3-Manifold invariants and periodicity of homology spheres (e-print, math.GT/9807011) | Zbl

[2] C. Blanchet; N. Habbeger; G. Masbaum; P. Vogel Three-Manifold invariants derived from the Kauffman bracket, Topology, Volume 31 (1992), pp. 685-699 | DOI | MR | Zbl

[3] H. Bass; J.W. Morgan The Smith conjecture, Pure App. Math., Volume 112 (1994) | MR | Zbl

[4] N. Chbili The Jones polynomials of freely periodic knots, J. Knot Th. Ram., Volume 9 (2000) no. 7, pp. 885-891 | DOI | MR | Zbl

[5] N. Chbili Le polynôme de Homfly des nœuds librement périodiques, C.R. Acad. Sci. Paris, série I, Volume 325 (1997), pp. 411-414 | Zbl

[6] D.L. Goldsmith Symmetric fibered links. Knots, groups and 3-manifolds, University press, 1975 | MR | Zbl

[7] J. Hoste; J. Przytycki A survey of skein modules of 3-manifolds, Proceeding of thel international conference on knot theory and related topics, Knots 90, Osaka (Japan) (1992), pp. 363-379 | Zbl

[8] L.H. Kauffman An invariant of regular isotopy, Trans. Amer. Math. Soc., Volume 318 (1990), pp. 417-471 | DOI | MR | Zbl

[9] W.B.R. Lickorish A representation of orientable combonatorial 3-manifolds, Ann. Math., Volume 76 (1962), pp. 531-540 | DOI | MR | Zbl

[10] W.B.R. Lickorish The skein method for 3-manifold invariants, J. Knot Th. Ram., Volume 2 (1993), pp. 171-194 | DOI | MR | Zbl

[12] K. Murasugi The Jones polynomials of periodic links, Pacific J. Math., Volume 131 (1988), pp. 319-329 | MR | Zbl

[13] V.V. Prasolov; A.B. Sossinsky Knots, Links, Braids and 3-manifolds, Trans. Math. Monographs, Vol. 154 | Zbl

[14] J. Przytycki; M. Sokolov Surgeries on periodic links and homology of periodic 3-manifolds (To appear in Math. Proc. Cambridge Phil. Soc., 131, Part 2) | MR | Zbl

[15] N. YU. Reshitikhin; V. Turaev Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math., Volume 103 (1991), pp. 547-597 | DOI | Zbl

[16] P. Traczyk Periodic knots and the skein polynomial, Invent. Math., Volume 106 (1991) no. 1, pp. 73-84 | DOI | MR | Zbl

[17] E. Witten Quantum field theory and the Jones polynomial, Comm. Math. Phys., Volume 121 (1989), pp. 351-399 | DOI | Zbl

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