Il est notoire que Sullivan a démontré que le groupe de difféotopie d’une variété de haute dimension simplement connexe est commensurable avec un groupe arithmétique, mais la signification du terme « commensurable » dans son théorème semble bien moins connue. Nous expliquons la raison pour laquelle ce résultat n’est plus vrai en utilisant la définition désomais standard de commensurabilité en exhibant une variété dont le groupe de difféotopie n’est pas résiduellement fini. Il n’est pas question d’un problème avec le théorème de Sullivan, mais plutôt d’y ajouter une glose.
It is well known that Sullivan showed that the mapping class group of a simply connected high-dimensional manifold is commensurable with an arithmetic group, but the meaning of “commensurable” in this statement seems to be less well known. We explain why this result fails with the now standard definition of commensurability by exhibiting a manifold whose mapping class group is not residually finite. We do not suggest any problem with Sullivan’s result: rather we provide a gloss for it.
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@article{CRMATH_2020__358_4_469_0,
author = {Krannich, Manuel and Randal-Williams, Oscar},
title = {Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic},
journal = {Comptes Rendus. Math\'ematique},
pages = {469--473},
publisher = {Acad\'emie des sciences, Paris},
volume = {358},
number = {4},
year = {2020},
doi = {10.5802/crmath.61},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.61/}
}
TY - JOUR AU - Krannich, Manuel AU - Randal-Williams, Oscar TI - Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic JO - Comptes Rendus. Mathématique PY - 2020 SP - 469 EP - 473 VL - 358 IS - 4 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.61/ DO - 10.5802/crmath.61 LA - en ID - CRMATH_2020__358_4_469_0 ER -
%0 Journal Article %A Krannich, Manuel %A Randal-Williams, Oscar %T Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic %J Comptes Rendus. Mathématique %D 2020 %P 469-473 %V 358 %N 4 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.61/ %R 10.5802/crmath.61 %G en %F CRMATH_2020__358_4_469_0
Krannich, Manuel; Randal-Williams, Oscar. Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic. Comptes Rendus. Mathématique, Tome 358 (2020) no. 4, pp. 469-473. doi : 10.5802/crmath.61. http://www.numdam.org/articles/10.5802/crmath.61/
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