logo CRAS
Comptes Rendus. Mathématique
Group Theory, Differential Topology
Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic
Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 469-473.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.61
Classification: 57R50,  11F06,  20E26
Manuel Krannich 1; Oscar Randal-Williams 1

1 Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
@article{CRMATH_2020__358_4_469_0,
     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},
}
TY  - JOUR
TI  - Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic
JO  - Comptes Rendus. Mathématique
PY  - 2020
DA  - 2020///
SP  - 469
EP  - 473
VL  - 358
IS  - 4
PB  - Académie des sciences, Paris
UR  - https://doi.org/10.5802/crmath.61
DO  - 10.5802/crmath.61
LA  - en
ID  - CRMATH_2020__358_4_469_0
ER  - 
%0 Journal Article
%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 https://doi.org/10.5802/crmath.61
%R 10.5802/crmath.61
%G en
%F CRMATH_2020__358_4_469_0
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/

[1] Kenneth S. Brown Cohomology of groups, Graduate Texts in Mathematics, 87, Springer, 1982, x+306 pages | MR: 672956 | Zbl: 0584.20036

[2] Pierre Deligne Extensions centrales non résiduellement finies de groupes arithmétiques, C. R. Math. Acad. Sci. Paris, Volume 287 (1978) no. 4, p. A203-A208 | MR: 507760 | Zbl: 0416.20042

[3] Søren Galatius; Oscar Randal-Williams Abelian quotients of mapping class groups of highly connected manifolds, Math. Ann., Volume 365 (2016) no. 1-2, pp. 857-879 | Article | MR: 3498929 | Zbl: 1343.55005

[4] Dennis Johnson; John J. Millson Modular Lagrangians and the theta multiplier, Invent. Math., Volume 100 (1990) no. 1, pp. 143-165 | Article | MR: 1037145 | Zbl: 0699.10042

[5] Michel A. Kervaire; John W. Milnor Groups of homotopy spheres. I, Ann. Math., Volume 77 (1963), pp. 504-537 | Article | MR: 148075 | Zbl: 0115.40505

[6] Manuel Krannich Mapping class groups of highly connected (4k+2)-manifolds (2019) (https://arxiv.org/abs/1902.10097)

[7] Matthias Kreck Isotopy classes of diffeomorphisms of (k-1)-connected almost-parallelizable 2k-manifolds, Algebraic topology, Aarhus 1978 (Lecture Notes in Mathematics), Volume 763 (1979), pp. 643-663 | Article | MR: 561244 | Zbl: 0421.57009

[8] Werner Meyer Die Signatur von lokalen Koeffizientensystemen und Faserbündeln, Bonn. Math. Schr. (1972) no. 53, p. viii+59 | MR: 305402 | Zbl: 0243.58004

[9] Werner Meyer Die Signatur von Flächenbündeln, Math. Ann., Volume 201 (1973), pp. 239-264 | Article | MR: 331382 | Zbl: 0241.55019

[10] Dave Witte Morris A lattice with no torsion-free subgroup of finite index (after P. Deligne) (2009) (Available at http://people.uleth.ca/~dave.morris/talks/deligne-torsion.pdf)

[11] Jean-Pierre Serre Arithmetic groups, Homological group theory (Proc. Sympos., Durham, 1977) (London Mathematical Society Lecture Note Series), Volume 36 (1979), pp. 105-136 | Article | MR: 564421 | Zbl: 0432.20042

[12] Dennis Sullivan Genetics of homotopy theory and the Adams conjecture, Ann. Math., Volume 100 (1974), pp. 1-79 | Article | MR: 442930 | Zbl: 0355.57007

[13] Dennis Sullivan Infinitesimal computations in topology, Publ. Math., Inst. Hautes Étud. Sci. (1977) no. 47, pp. 269-331 | Article | Numdam | MR: 0646078 | Zbl: 0374.57002

[14] René Thom Quelques propriétés globales des variétés différentiables, Comment. Math. Helv., Volume 28 (1954), pp. 17-86 | Article | MR: 61823 | Zbl: 0057.15502

Cited by Sources: