Nous démontrons le résultat suivant : si les invariants de Vassiliev distinguent les nœuds dans toute sphère d'homotopie, alors la conjecture de Poincaré est vraie, c'est-à-dire toute sphère d'homotopie est homéomorphe à la sphère standard. D'un autre coté, dans toute variété de Whitehead il existe des nœuds qui ne sont pas distingués par les invariants de Vassiliev.
We prove the following result: if Vassiliev invariants distinguish knots in each homotopy sphere, then the Poincaré conjecture is true, in other words every homotopy sphere is homeomorphic to the standard sphere. On the other hand, in every Whitehead manifold there exist knots that cannot be distinguished by Vassiliev invariants.
Accepté le :
Publié le :
@article{CRMATH_2002__334_11_1005_0,
author = {Eisermann, Michael},
title = {Invariants de {Vassiliev} et conjecture de {Poincar\'e}},
journal = {Comptes Rendus. Math\'ematique},
pages = {1005--1010},
publisher = {Elsevier},
volume = {334},
number = {11},
year = {2002},
doi = {10.1016/S1631-073X(02)02375-0},
language = {fr},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02375-0/}
}
TY - JOUR AU - Eisermann, Michael TI - Invariants de Vassiliev et conjecture de Poincaré JO - Comptes Rendus. Mathématique PY - 2002 SP - 1005 EP - 1010 VL - 334 IS - 11 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02375-0/ DO - 10.1016/S1631-073X(02)02375-0 LA - fr ID - CRMATH_2002__334_11_1005_0 ER -
%0 Journal Article %A Eisermann, Michael %T Invariants de Vassiliev et conjecture de Poincaré %J Comptes Rendus. Mathématique %D 2002 %P 1005-1010 %V 334 %N 11 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02375-0/ %R 10.1016/S1631-073X(02)02375-0 %G fr %F CRMATH_2002__334_11_1005_0
Eisermann, Michael. Invariants de Vassiliev et conjecture de Poincaré. Comptes Rendus. Mathématique, Tome 334 (2002) no. 11, pp. 1005-1010. doi : 10.1016/S1631-073X(02)02375-0. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02375-0/
[1] On the Vassiliev knot invariants, Topology, Volume 34 (1995), pp. 423-472
[2] Necessary and sufficient conditions a 3-manifold be , Ann. of Math., Volume 68 (1958), pp. 17-37
[3] Knot polynomials and Vassiliev's invariants, Invent. Math., Volume 111 (1993), pp. 225-270
[4] A characterisation of punctured open 3-cells, Proc. Amer. Math. Soc., Volume 28 (1971), pp. 295-298
[5] Differential Topology, Graduate Texts in Math., 33, Springer-Verlag, New York, 1976
[6] Locally Euclidean factors of E4 which cannot be imbedded in E3, Ann. of Math., Volume 76 (1962), pp. 541-546
[7] Finite type link invariants of 3-manifolds, Topology, Volume 33 (1994), pp. 45-71
[8] Some contractible open 3-manifolds, Trans. Amer. Math. Soc., Volume 102 (1962), pp. 373-382
[9] A certain open manifold whose group is unity, Quart. J. Math., Volume 6 (1935), pp. 268-279
Cité par Sources :
