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, p. 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 : https://doi.org/10.5802/aif.1848
Classification:  57M27
Keywords: periodic 3-manifold, periodic link, homology sphere, quantum invariants
@article{AIF_2001__51_4_1135_0,
     author = {Chbili, Nafaa},
     title = {Les invariants $\theta \_p$ des 3-vari\'et\'es p\'eriodiques},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {4},
     year = {2001},
     pages = {1135-1150},
     doi = {10.5802/aif.1848},
     zbl = {0997.57031},
     mrnumber = {1849218},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2001__51_4_1135_0}
}
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. http://www.numdam.org/item/AIF_2001__51_4_1135_0/

[MR] G. Masbaum; J. Roberts A simple proof of integrality of quantum invariants at prime roots of unity, Math. Proc. Cambridge. Phil. Soc., Tome 121 (1997), pp. 443-454 | Article | MR 1434653 | Zbl 0882.57010

[1] J.K. Bartoszynska; J. Przytycki 3-Manifold invariants and periodicity of homology spheres (e-print, math.GT/9807011) | Zbl 1008.57013

[2] C. Blanchet; N. Habbeger; G. Masbaum; P. Vogel Three-Manifold invariants derived from the Kauffman bracket, Topology, Tome 31 (1992), pp. 685-699 | Article | MR 1191373 | Zbl 0771.57004

[3] H. Bass; J.W. Morgan The Smith conjecture, Pure App. Math., Tome 112 (1994) | MR 758460 | Zbl 0599.57001

[4] N. Chbili The Jones polynomials of freely periodic knots, J. Knot Th. Ram., Tome 9 (2000) no. 7, pp. 885-891 | Article | MR 1780593 | Zbl 0999.57012

[5] N. Chbili Le polynôme de Homfly des nœuds librement périodiques, C.R. Acad. Sci. Paris, série I, Tome 325 (1997), pp. 411-414 | Zbl 0884.57008

[6] D.L. Goldsmith Symmetric fibered links. Knots, groups and 3-manifolds, University press (1975) | MR 380766 | Zbl 0331.55001

[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), Walter de Gruyter (1992), pp. 363-379 | Zbl 0772.57022

[8] L.H. Kauffman An invariant of regular isotopy, Trans. Amer. Math. Soc., Tome 318 (1990), pp. 417-471 | Article | MR 958895 | Zbl 0763.57004

[9] W.B.R. Lickorish A representation of orientable combonatorial 3-manifolds, Ann. Math., Tome 76 (1962), pp. 531-540 | Article | MR 151948 | Zbl 0106.37102

[10] W.B.R. Lickorish The skein method for 3-manifold invariants, J. Knot Th. Ram., Tome 2 (1993), pp. 171-194 | Article | MR 1227009 | Zbl 0793.57003

[12] K. Murasugi The Jones polynomials of periodic links, Pacific J. Math., Tome 131 (1988), pp. 319-329 | MR 922222 | Zbl 0661.57001

[13] V.V. Prasolov; A.B. Sossinsky Knots, Links, Braids and 3-manifolds, Trans. Math. Monographs, Tome Vol. 154 | Zbl 0864.57002

[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 1857121 | Zbl 0985.57013

[15] N. Yu. Reshitikhin; V. Turaev Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math., Tome 103 (1991), pp. 547-597 | Article | Zbl 0725.57007

[16] P. Traczyk Periodic knots and the skein polynomial, Invent. Math., Tome 106 (1991) no. 1, pp. 73-84 | Article | MR 1123374 | Zbl 0753.57008

[17] E. Witten Quantum field theory and the Jones polynomial, Comm. Math. Phys., Tome 121 (1989), pp. 351-399 | Article | Zbl 0667.57005