The rational stable homology of mapping class groups of universal nil-manifolds

Szymik, Markus

Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 783-803.

Nous calculons l’homologie rationnelle stable des groupes d’automorphismes de groupes nilpotents libres. Ces groupes s’intercalent entre les groupes généraux linéaires sur l’anneau des entiers et les groupes d’automorphismes de groupes libres, et nous employons l’homologie de foncteurs pour nous réduire au cas abélien. A titre d’application, nous calculons également l’homologie rationnelle stable des groupes d’automorphismes extérieurs et des groupes modulaires des variétés asphériques associées dans les catégories TOP, PL, et DIFF.

We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute the rational stable homology of the outer automorphism groups and of the mapping class groups of the associated aspherical nil-manifolds in the TOP, PL, and DIFF categories.

Classification : 20J05, 20E36, 19M05, 18A25, 18G40
Keywords: stable homology, automorphism groups, nilpotent groups, functor categories, Hochschild homology, stable K-theory, spectral sequences
Mot clés : homologie stable, groupes d’automorphismes, groupes nilpotents, catégories de foncteurs, homologie de Hochschild, K-théorie stable, suites spectrales

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