Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic

Manuel Krannich; Oscar Randal-Williams

doi:10.5802/crmath.61

Théorie des groupes, Topologie différentielle

Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic

Manuel Krannich ¹ ; Oscar Randal-Williams ¹

¹ Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 469-473.

Résumé
Abstract

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.

Reçu le : 2019-10-25
Révisé le : 2020-03-17
Accepté le : 2020-04-26
Publié le : 2020-07-28

DOI : 10.5802/crmath.61

Classification : 57R50, 11F06, 20E26

Affiliations des auteurs :

Manuel Krannich ¹ ; Oscar Randal-Williams ¹

¹ Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Manuel Krannich and Oscar Randal-Williams},
     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},
}

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Manuel Krannich; Oscar Randal-Williams. Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 469-473. doi : 10.5802/crmath.61. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.61/

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