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

Turchin, Victor; Willwacher, Thomas

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

On montre que l'homologie de Hochschild-Pirashvili sur toute suspension admet une certaine décomposition de Hodge. Pour toute application entre suspensions $f:\Sigma Y\to \Sigma Z$, 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 $\mathrm{Out}\left(F_n\right)$ qui ne se factorisent pas en général par $\mathrm{GL}(n,\mathbb{Z})$. Les représentations ainsi obtenues sont naturellement filtrées de façon à ce que l'action sur les quotients gradués se factorise par $\mathrm{GL}(n,\mathbb{Z})$.

We show that the Hochschild-Pirashvili homology on any suspension admits the so called Hodge splitting. For a map between suspensions $f:\Sigma Y\to \Sigma Z$, 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 $\mathrm{Out}\left(F_n\right)$ that do not factor in general through $\mathrm{GL}(n,\mathbb{Z})$. The obtained representations are naturally filtered in such a way that the action on the graded quotients does factor through $\mathrm{GL}(n,\mathbb{Z})$.

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

Ayala, D.; Francis, J. Factorization homology of topological manifolds, J. Topol., Volume 8 (2015), pp. 1045-1084 (ISSN: 1753-8416) | DOI | MR | Zbl

Arone, G.; Turchin, V. On the rational homology of high-dimensional analogues of spaces of long knots, Geom. Topol., Volume 18 (2014), pp. 1261-1322 (ISSN: 1465-3060) | DOI | MR | Zbl

Arone, G.; Turchin, V. Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots, Ann. Inst. Fourier, Volume 65 (2015), pp. 1-62 http://aif.cedram.org/item?id=AIF_2015__65_1_1_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Buijs, U.; Félix, Y.; Murillo, A. $L_{\infty }$ models of based mapping spaces, J. Math. Soc. Japan, Volume 63 (2011), pp. 503-524 http://projecteuclid.org/euclid.jmsj/1303737796 (ISSN: 0025-5645) | MR | Zbl

Block, J.; Lazarev, A. André-Quillen cohomology and rational homotopy of function spaces, Adv. Math., Volume 193 (2005), pp. 18-39 (ISSN: 0001-8708) | DOI | MR | Zbl

Brown, E. H. J.; Szczarba, R. H. On the rational homotopy type of function spaces, Trans. Amer. Math. Soc., Volume 349 (1997), pp. 4931-4951 (ISSN: 0002-9947) | DOI | MR | Zbl

Bridson, M. R.; Vogtmann, K. Abelian covers of graphs and maps between outer automorphism groups of free groups, Math. Ann., Volume 353 (2012), pp. 1069-1102 (ISSN: 0025-5831) | DOI | MR | Zbl

Carlsson, G.; Douglas, C. L.; Dundas, B. I. Higher topological cyclic homology and the Segal conjecture for tori, Adv. Math., Volume 226 (2011), pp. 1823-1874 (ISSN: 0001-8708) | DOI | MR | Zbl

Djament, A.; Vespa, C. Sur l'homologie des groupes d'automorphismes des groupes libres à coefficients polynomiaux, Comment. Math. Helv., Volume 90 (2015), pp. 33-58 (ISSN: 0010-2571) | DOI | MR | Zbl

Félix, Y.; Halperin, S.; Thomas, J.-C., Graduate Texts in Math., 205, Springer, 2001, 535 pages (ISBN: 0-387-95068-0) | DOI | MR | Zbl

Félix, Y.; Halperin, S.; Thomas, J.-C., World Scientific Publishing Co. Pte. Ltd., 2015, 412 pages (ISBN: 978-981-4651-42-4) | DOI | MR

Fresse, B., Lecture Notes in Math., 1967, Springer, 2009, 308 pages (ISBN: 978-3-540-89055-3) | DOI | MR | Zbl

Ginot, G., Mathematical aspects of quantum field theories (Math. Phys. Stud.), Springer, 2015, pp. 429-552 | DOI | MR | Zbl

Grunewald, F.; Lubotzky, A. Linear representations of the automorphism group of a free group, Geom. Funct. Anal., Volume 18 (2009), pp. 1564-1608 (ISSN: 1016-443X) | DOI | MR | Zbl

Gerstenhaber, M.; Schack, S. D. A Hodge-type decomposition for commutative algebra cohomology, J. Pure Appl. Algebra, Volume 48 (1987), pp. 229-247 (ISSN: 0022-4049) | DOI | MR | Zbl

Ginot, G.; Tradler, T.; Zeinalian, M. A Chen model for mapping spaces and the surface product, Ann. Sci. Éc. Norm. Supér., Volume 43 (2010), pp. 811-881 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Ginot, G.; Tradler, T.; Zeinalian, M. Higher Hochschild homology, topological chiral homology and factorization algebras, Comm. Math. Phys., Volume 326 (2014), pp. 635-686 (ISSN: 0010-3616) | DOI | MR | Zbl

Intermont, M.; Johnson, B.; McCarthy, R. The rank filtration and Robinson's complex, J. Pure Appl. Algebra, Volume 212 (2008), pp. 735-752 (ISSN: 0022-4049) | DOI | MR | Zbl

Kielak, D. Outer automorphism groups of free groups: linear and free representations, J. Lond. Math. Soc., Volume 87 (2013), pp. 917-942 (ISSN: 0024-6107) | DOI | MR | Zbl

Kielak, D. Low-dimensional free and linear representations of $\mathrm{Out}\left(F_3\right)$ , J. Group Theory, Volume 18 (2015), pp. 913-949 (ISSN: 1433-5883) | DOI | MR | Zbl

Loday, J.-L. Opérations sur l'homologie cyclique des algèbres commutatives, Invent. math., Volume 96 (1989), pp. 205-230 (ISSN: 0020-9910) | DOI | MR | Zbl

Lurie, J., Current developments in mathematics, 2008, Int. Press, 2009, pp. 129-280 | MR | Zbl

Loday, J.-L.; Vallette, B., Grundl. math. Wiss., 346, Springer, 2012, 634 pages (ISBN: 978-3-642-30361-6) | DOI | MR | Zbl

Patras, F. Generic algebras and iterated Hochschild homology, J. Pure Appl. Algebra, Volume 162 (2001), pp. 337-357 (ISSN: 0022-4049) | DOI | MR | Zbl

Patras, F. La décomposition en poids des algèbres de Hopf, Ann. Inst. Fourier, Volume 43 (1993), pp. 1067-1087 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Patras, F. L'algèbre des descentes d'une bigèbre graduée, J. Algebra, Volume 170 (1994), pp. 547-566 (ISSN: 0021-8693) | DOI | MR | Zbl

Pirashvili, T. Dold-Kan type theorem for $\Gamma$-groups, Math. Ann., Volume 318 (2000), pp. 277-298 (ISSN: 0025-5831) | DOI | MR | Zbl

Pirashvili, T. Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup., Volume 33 (2000), pp. 151-179 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Patras, F.; Thomas, J.-C. Cochain algebras of mapping spaces and finite group actions, Topology Appl., Volume 128 (2003), pp. 189-207 (ISSN: 0166-8641) | DOI | MR | Zbl

Robert-Nicoud, D. Deformation theory with homotopy algebra structures on tensor products, Doc. Math., Volume 23 (2018), pp. 189-240 (ISSN: 1431-0635) | DOI | MR | Zbl

Schlichtkrull, C. Higher topological Hochschild homology of Thom spectra, J. Topol., Volume 4 (2011), pp. 161-189 (ISSN: 1753-8416) | DOI | MR | Zbl

Songhafouo Tsopméné, P. A. The rational homology of spaces of long links, Algebr. Geom. Topol., Volume 16 (2016), pp. 757-782 (ISSN: 1472-2747) | DOI | MR | Zbl

Songhafouo Tsopméné, P. A.; Turchin, V. Rational homology and homotopy of high-dimensional string links, Forum Math., Volume 30 (2018), pp. 1209-1235 (ISSN: 0933-7741) | DOI | MR | Zbl

Turchin, V. Hodge-type decomposition in the homology of long knots, J. Topol., Volume 3 (2010), pp. 487-534 (ISSN: 1753-8416) | DOI | MR | Zbl

Turchin, V.; Willwacher, T. Commutative hairy graphs and representations of $\mathrm{Out}\left(F_r\right)$ , J. Topol., Volume 10 (2017), pp. 386-411 (ISSN: 1753-8416) | DOI | MR | Zbl

Turchin, V.; Willwacher, T. Relative (non-)formality of the little cubes operads and the algebraic Cerf lemma, Amer. J. Math., Volume 140 (2018), pp. 277-316 (ISSN: 0002-9327) | DOI | MR | Zbl

Weibel, C. A., Cambridge Studies in Advanced Math., 38, Cambridge Univ. Press, 1994, 450 pages (ISBN: 0-521-43500-5; 0-521-55987-1) | DOI | MR | Zbl

Weiss, M. Embeddings from the point of view of immersion theory. I, Geom. Topol., Volume 3 (1999), pp. 67-101 (ISSN: 1465-3060) | DOI | MR | Zbl