Configuration spaces on a wedge of spheres and Hochschild–Pirashvili homology

Gadish, Nir; Hainaut, Louis

doi:10.5802/ahl.213

Configuration spaces on a wedge of spheres and Hochschild–Pirashvili homology
[Espaces de configuration sur un bouquet de sphères et homologie de Hochschild–Pirashvili]

Gadish, Nir ¹ ; Hainaut, Louis ²

¹Department of Mathematics, University of Michigan, 530 Church St, Ann Arbor, MI 48109 (USA)
²Stockholm University, Department of Mathematics, 106 91 Stockholm (Sweden)

Annales Henri Lebesgue, Tome 7 (2024), pp. 841-902

Abstract (VO)
Résumé

We study the compactly supported rational cohomology of configuration spaces of points on wedges of spheres, equipped with natural actions of the symmetric group and the group $Out (F_{g})$ of outer automorphisms of the free group. These representations show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups and Hochschild–Pirashvili cohomology.

We show that these cohomology representations form a polynomial functor, and use various geometric models to compute many of its composition factors. We further compute the composition factors completely for all configurations of $n \leq 10$ points. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight $0$ component of $H_{c}^{*} (ℳ_{2, n})$ .

Nous étudions la cohomologie rationnelle à support compact des espaces de configurations de points sur les bouquets de sphères, équipée d’actions naturelles du groupe symétrique et du groupe $Out (F_{g})$ des automorphismes extérieurs du groupe libre. Ces représentations apparaissent dans des parties qui ne semblent pas avoir de lien entre elles, notamment la cohomologie des espaces de modules de courbes ainsi que les foncteurs polynomiaux sur les groupes libres et la cohomologie de Hochschild–Pirashvili.

Nous prouvons que ces représentations obtenues par la cohomologie ont une structure de foncteur polynomial et utilisons divers modèles géométriques pour calculer une grande quantité de leurs facteurs de composition. De plus, nous calculons complètement les facteurs de composition pour toutes les configurations de $n \leq 10$ points. Comme application de cette analyse, nous obtenons une nouvelle borne inférieure à croissance super-exponentielle pour l’action du groupe symétrique sur la composante de poids $0$ de $H_{c}^{*} (ℳ_{2, n})$ .

Reçu le : 2023-04-21
Accepté le : 2024-05-28
Publié le : 2024-09-05

MR Zbl

DOI : 10.5802/ahl.213

Classification : 55R80, 14H10, 14Q05, 13D03
Keywords: configuration spaces, polynomial functors, moduli spaces

Affiliations des auteurs :

Gadish, Nir ¹ ; Hainaut, Louis ²

¹ Department of Mathematics, University of Michigan, 530 Church St, Ann Arbor, MI 48109 (USA)
² Stockholm University, Department of Mathematics, 106 91 Stockholm (Sweden)

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AHL_2024__7__841_0,
     author = {Gadish, Nir and Hainaut, Louis},
     title = {Configuration spaces on a wedge of spheres and {Hochschild{\textendash}Pirashvili} homology},
     journal = {Annales Henri Lebesgue},
     pages = {841--902},
     year = {2024},
     publisher = {\'ENS Rennes},
     volume = {7},
     doi = {10.5802/ahl.213},
     mrnumber = {4799912},
     zbl = {07914808},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/ahl.213/}
}

TY  - JOUR
AU  - Gadish, Nir
AU  - Hainaut, Louis
TI  - Configuration spaces on a wedge of spheres and Hochschild–Pirashvili homology
JO  - Annales Henri Lebesgue
PY  - 2024
SP  - 841
EP  - 902
VL  - 7
PB  - ÉNS Rennes
UR  - https://www.numdam.org/articles/10.5802/ahl.213/
DO  - 10.5802/ahl.213
LA  - en
ID  - AHL_2024__7__841_0
ER  -

%0 Journal Article
%A Gadish, Nir
%A Hainaut, Louis
%T Configuration spaces on a wedge of spheres and Hochschild–Pirashvili homology
%J Annales Henri Lebesgue
%D 2024
%P 841-902
%V 7
%I ÉNS Rennes
%U https://www.numdam.org/articles/10.5802/ahl.213/
%R 10.5802/ahl.213
%G en
%F AHL_2024__7__841_0

Gadish, Nir; Hainaut, Louis. Configuration spaces on a wedge of spheres and Hochschild–Pirashvili homology. Annales Henri Lebesgue, Tome 7 (2024), pp. 841-902. doi: 10.5802/ahl.213

Bibliographie
Cité par

[Ada97] Adamchik, Victor On Stirling numbers and Euler sums, J. Comput. Appl. Math., Volume 79 (1997) no. 1, pp. 119-130 | DOI | Zbl | MR

[AT14] Arone, Gregory; Turchin, Victor On the rational homology of high-dimensional analogues of spaces of long knots, Geom. Topol., Volume 18 (2014) no. 3, pp. 1261-1322 | DOI | Zbl | MR

[BBP23] Brendle, Tara; Broaddus, Nathan; Putman, Andrew The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary, Isr. J. Math. (2023), pp. 1-38 | DOI | Zbl | MR

[BCGY23] Bibby, Christin; Chan, Melody; Gadish, Nir; Yun, Claudia He Homology Representations of Compactified Configurations on Graphs Applied to $ℳ_{2, n}$ , Exp. Math. (2023), pp. 1-13 | DOI | MR | Zbl

[BG23] Bibby, Christin; Gadish, Nir A generating function approach to new representation stability phenomena in orbit configuration spaces, Trans. Am. Math. Soc., Ser. B, Volume 10 (2023) no. 9, pp. 241-287 | DOI | MR | Zbl

[CFGP23] Chan, Melody; Faber, Carel; Galatius, Søren; Payne, Sam The $S_{n}$ -equivariant top weight Euler characteristic of $ℳ_{g, n}$ , Am. J. Math., Volume 145 (2023) no. 5, pp. 1549-1585 | DOI | Zbl | MR

[CGP22] Chan, Melody; Galatius, Søren; Payne, Sam Topology of Moduli Spaces of Tropical Curves with Marked Points, Facets of Algebraic Geometry: A Collection in Honor of William Fulton’s 80th Birthday (London Mathematical Society Lecture Note Series), Volume 1, Cambridge University Press, 2022, pp. 77-131 | DOI | Zbl

[Cha21] Chan, Melody Topology of the tropical moduli spaces $Δ_{2, n}$ , Beitr. Algebra Geom., Volume 63 (2021) no. 1, pp. 69-93 | DOI | Zbl | MR

[Coh76] Cohen, Frederick R. The homology of $C_{n + 1}$ -Spaces, $n \geq 0$ , The Homology of Iterated Loop Spaces (Lecture Notes in Mathematics), Volume 533, Springer, 1976, pp. 207-351 | DOI

[CT78] Cohen, Frederick R.; Taylor, Laurence R. Computations of Gelfand–Fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, Geometric applications of homotopy theory I, Springer, 1978, pp. 106-143 | Zbl | DOI

[DV15] Djament, Aurélien; Vespa, Christine Sur l’homologie des groupes d’automorphismes des groupes libres à coefficients polynomiaux, Comment. Math. Helv., Volume 90 (2015) no. 1, pp. 33-58 | DOI | Zbl | MR

[EM54] Eilenberg, Samuel; MacLane, Saunders On the Groups $H (Π, n)$ , II: Methods of Computation, Ann. Math., Volume 60 (1954) no. 1, pp. 49-139 | DOI | Zbl

[ER19] Early, Nicholas; Reiner, Victor On configuration spaces and Whitehouse’s lifts of the Eulerian representations, J. Pure Appl. Algebra, Volume 223 (2019) no. 10, pp. 4524-4535 | DOI | Zbl | MR

[Erd53] Erdös, Paul On a conjecture of Hammersley, J. Lond. Math. Soc., Volume 1 (1953) no. 2, pp. 232-236 | DOI | Zbl | MR

[FZ00] Feichtner, Eva Maria; Ziegler, Günter M. The integral cohomology algebras of ordered configuration spaces of spheres, Doc. Math., Volume 5 (2000), pp. 115-139 | Zbl | MR | DOI

[Get95] Getzler, Ezra Operads and Moduli Spaces of Genus 0 Riemann Surfaces, The Moduli Space of Curves (Dijkgraaf, Robbert H.; Faber, Carel F.; van der Geer, Gerard B. M., eds.), Birkhäuser (1995), pp. 199-230 | DOI | Zbl

[Get99a] Getzler, Ezra The homology groups of some two-step nilpotent Lie algebras associated to symplectic vector spaces (1999) | arXiv

[Get99b] Getzler, Ezra Resolving mixed Hodge modules on configuration spaces, Duke Math. J., Volume 96 (1999) no. 1, pp. 175-203 | DOI | Zbl | MR

[GK98] Getzler, Ezra; Kapranov, Mikhail M. Modular operads, Compos. Math., Volume 110 (1998) no. 1, pp. 65-125 | DOI | Zbl | MR

[GK02] Ghrist, Robert W.; Koditschek, Daniel E. Safe cooperative robot dynamics on graphs, SIAM J. Control Optim., Volume 40 (2002) no. 5, pp. 1556-1575 | DOI | Zbl | MR

[HP23] Hainaut, Louis; Petersen, Dan Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies (2023) | arXiv

[HPV15] Hartl, Manfred; Pirashvili, Teimuraz; Vespa, Christine Polynomial functors from algebras over a set-operad and nonlinear Mackey functors, Int. Math. Res. Not., Volume 2015 (2015) no. 6, pp. 1461-1554 | DOI | Zbl | MR

[Hyd20] Hyde, Trevor Polynomial Factorization Statistics and Point Configurations in $ℝ^{3}$ , Int. Math. Res. Not., Volume 2020 (2020) no. 24, pp. 10154-10179 | DOI | Zbl | MR

[HZ86] Harer, John; Zagier, Don The Euler characteristic of the moduli space of curves., Invent. Math., Volume 85 (1986), pp. 457-485 | DOI | Zbl | MR

[Hô17] Hô, Quoc P. Free factorization algebras and homology of configuration spaces in algebraic geometry, Sel. Math., New Ser., Volume 23 (2017) no. 4, pp. 2437-2489 | DOI | Zbl | MR

[Hô20] Hô, Quoc P. Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras (2020) | arXiv

[Joy86] Joyal, André Foncteurs analytiques et espèces de structures, Combinatoire énumérative (Labelle, Gilbert; Leroux, Pierre, eds.) (Lecture Notes in Mathematics), Volume 1234, Springer (1986), pp. 126-159 | DOI | Zbl

[JR11] Jiménez Rolland, Rita Representation stability for the cohomology of the moduli space $ℳ_{g}^{n}$ , Algebr. Geom. Topol., Volume 11 (2011) no. 5, pp. 3011-3041 | DOI | Zbl | MR

[Kri94] Kriz, Igor On the Rational Homotopy Type of Configuration Spaces, Ann. Math., Volume 139 (1994) no. 2, pp. 227-237 | DOI | Zbl

[LS05] Longoni, Riccardo; Salvatore, Paolo Configuration spaces are not homotopy invariant, Topology, Volume 44 (2005) no. 2, pp. 375-380 | DOI | Zbl | MR

[Mac95] Macdonald, Ian G. Symmetric functions and Hall polynomials, Oxford University Press, 1995 | Zbl | MR | DOI

[Nie17] Nielsen, J. Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden, Math. Ann., Volume 78 (1917) no. 1-4, pp. 385-397 | DOI | Zbl | MR

[Pag23] Pagaria, Roberto The Frobenius characteristic of the Orlik-Terao algebra of type A, Int. Math. Res. Not. (2023) no. 13, pp. 11577-11591 | DOI | Zbl | MR

[Pet13] Petersen, Dan Cohomology of local systems on loci of d-elliptic abelian surfaces, Mich. Math. J., Volume 62 (2013) no. 4, pp. 705-720 | DOI | Zbl | MR

[Pet15] Petersen, Dan Cohomology of local systems on the moduli of principally polarized abelian surfaces, Pac. J. Math., Volume 275 (2015) no. 1, pp. 39-61 | MR | DOI | Zbl

[Pet17] Petersen, Dan A spectral sequence for stratified spaces and configuration spaces of points, Geom. Topol., Volume 21 (2017) no. 4, pp. 2527-2555 | MR | DOI | Zbl

[Pet20] Petersen, Dan Cohomology of generalized configuration spaces, Compos. Math., Volume 156 (2020) no. 2, pp. 251-298 | DOI | Zbl | MR

[Pir00] Pirashvili, Teimuraz Hodge decomposition for higher order Hochschild homology, Ann. Sci. Éc. Norm. Supér., Volume 33 (2000) no. 2, pp. 151-179 | DOI | Zbl | MR | Numdam

[Pow21] Powell, Geoffrey On analytic contravariant functors on free groups (2021) | arXiv

[Pow22] Powell, Geoffrey Baby bead representations (2022) | arXiv

[Pow24] Powell, Geoffrey Outer functors and a general operadic framework, J. Algebra, Volume 644 (2024), pp. 526-562 | DOI | MR | Zbl

[PT21] Petersen, Dan; Tosteson, Phil Factorization statistics and bug-eyed configuration spaces, Geom. Topol., Volume 25 (2021) no. 7, pp. 3691-3723 | DOI | Zbl | MR

[PV18] Powell, Geoffrey; Vespa, Christine Higher Hochschild homology and exponential functors (2018) | arXiv

[Sta12] Stanley, Richard P. Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, 1, Cambridge University Press, 2012 | DOI | Zbl

[Tot96] Totaro, Burt Configuration spaces of algebraic varieties, Topology, Volume 35 (1996) no. 4, pp. 1057-1067 | DOI | Zbl | MR

[TSJ + 20] The Sage Developers; Stein, William; Joyner, David; Kohel, David; Cremona, John; Eröcal, Burçin SageMath, version 9.3, 2020 (http://www.sagemath.org)

[TW17] Turchin, Victor; Willwacher, Thomas Commutative hairy graphs and representations of $Out (F_{r})$ , J. Topol., Volume 10 (2017) no. 2, pp. 386-411 | DOI | Zbl | MR

[TW19] Turchin, Victor; Willwacher, Thomas Hochschild-Pirashvili homology on suspensions and representations of $Out (F_{n})$ , Ann. Sci. Éc. Norm. Supér., Volume 52 (2019) no. 3, pp. 761-795 | DOI | Zbl | MR | Numdam

[Ves22] Vespa, Christine On the functors associated with beaded open Jacobi diagrams (2022) | arXiv

Cité par Sources :