Hochschild-Pirashvili homology  on suspensions  and representations of $\mathrm {Out}(F_n)$

Turchin, Victor; Willwacher, Thomas

doi:10.24033/asens.2396

Hochschild-Pirashvili homology on suspensions and representations of

Out (F_{n})

[Homologie de Hochschild-Pirashvili sur les suspensions et représentations de

Out (F_{n})

]

Turchin, Victor ; Willwacher, Thomas

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 3, pp. 761-795.

Résumé
Abstract

On montre que l'homologie de Hochschild-Pirashvili sur toute suspension admet une certaine décomposition de Hodge. Pour toute application entre suspensions $f : Σ Y \to Σ Z$ , l'application induite en homologie de Hochschild-Pirashvili préserve cette décomposition si $f$ est une suspension. Dans le cas contraire, on montre que la décomposition est préservée uniquement en tant que filtration. Dans le cas particulier d'un bouquet de cercles, l'homologie de Hochschild-Pirashvili produit de nouvelles représentations de $Out (F_{n})$ qui ne se factorisent pas en général par $GL (n, ℤ)$ . Les représentations ainsi obtenues sont naturellement filtrées de façon à ce que l'action sur les quotients gradués se factorise par $GL (n, ℤ)$ .

We show that the Hochschild-Pirashvili homology on any suspension admits the so called Hodge splitting. For a map between suspensions $f : Σ Y \to Σ Z$ , the induced map in the Hochschild-Pirashvili homology preserves this splitting if $f$ is a suspension. If $f$ is not a suspension, we show that the splitting is preserved only as a filtration. As a special case, we obtain that the Hochschild-Pirashvili homology on wedges of circles produces new representations of $Out (F_{n})$ that do not factor in general through $GL (n, ℤ)$ . The obtained representations are naturally filtered in such a way that the action on the graded quotients does factor through $GL (n, ℤ)$ .

Publié le : 2019-11-12

MR Zbl

DOI : 10.24033/asens.2396

Classification : 55N99, 19D55, 13D03, 20C10.
Keywords: Higher Hochschild homology, Hodge decomposition, outer automorphism group of a free group, Poincaré-Birkhoff-Witt filtration, commutative-Lie Koszul duality.
Mot clés : Homologie de Hochschild supérieure, décomposition de Hodge, groupe d'automorphismes extérieurs d'un groupe libre, filtration de Poincaré-Birkhoff-Witt, dualité de Koszul commutative-Lie

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     author = {Turchin, Victor and Willwacher, Thomas},
     title = {Hochschild-Pirashvili homology  on suspensions  and representations of~$\mathrm {Out}(F_n)$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {761--795},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
     number = {3},
     year = {2019},
     doi = {10.24033/asens.2396},
     mrnumber = {3982870},
     zbl = {1435.55005},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2396/}
}

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Turchin, Victor; Willwacher, Thomas. Hochschild-Pirashvili homology  on suspensions  and representations of $\mathrm {Out}(F_n)$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 3, pp. 761-795. doi : 10.24033/asens.2396. https://www.numdam.org/articles/10.24033/asens.2396/

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