A direct computation of the cohomology of the braces operad

Vasily Dolgushev 1  and Thomas Willwacher 2
  • 1 Department of Mathematics, Temple University, Wachman Hall , Rm. 638, 1805 N. Broad St., Philadelphia, PA, 19122, USA
  • 2 Institute of Mathematics, University of Zürich, Winterthurerstraße 190, 8057 Zürich, Switzerland
Vasily Dolgushev and Thomas Willwacher

Abstract

We give a self-contained and purely combinatorial proof of the well-known fact that the cohomology of the braces operad is the operad 𝖦𝖾𝗋 governing Gerstenhaber algebras.

  • [1]

    Berger C. and Fresse B., Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 1, 135–174.

  • [2]

    Calaque D. and Willwacher T., Triviality of the higher formality theorem, Proc. Amer. Math. Soc. 143 (2015), no. 12, 5181–5193.

  • [3]

    Deligne P., Letter to V. Drinfeld, M. Gerstenhaber, J. P. May, V. Schechtman and J. Stasheff, unpublished, 1993.

  • [4]

    Dolgushev V. A. and Rogers C. L., Notes on algebraic operads, graph complexes, and Willwacher’s construction, Mathematical Aspects of Quantization, Contemp. Math. 583, American Mathematical Society, Providence (2012), 25–145.

  • [5]

    Dolgushev V. A. and Willwacher T. H., Operadic Twisting – with an application to Deligne’s conjecture, J. Pure Appl. Algebra 219 (2015), no. 5, 1349–1428.

  • [6]

    Drinfeld V. G., On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal ( ¯ / ) ${{\rm Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$ (in Russian), Algebra i Analiz 2 (1990), no. 4, 149–181.

  • [7]

    Gerstenhaber M., The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288.

  • [8]

    Getzler E., Cartan homotopy formulas and the Gauss–Manin connection in cyclic homology, Quantum Deformations of Algebras and Their Representations (Rehovot 1992), Israel Math. Conf. Proc. 7, Bar-Ilan University, Ramat Gan (1993), 65–78.

  • [9]

    Getzler E. and Jones J. D. S., Operads, homotopy algebra and iterated integrals for double loop spaces, preprint 1994, https://arxiv.org/abs/hep-th/9403055.

  • [10]

    Kaufmann R. M., Operads, moduli of surfaces and quantum algebras, Woods Hole Mathematics, Ser. Knots Everything 34, World Scientific, Hackensack (2004), 133–224.

  • [11]

    Kaufmann R. M., On several varieties of cacti and their relations, Algebr. Geom. Topol. 5 (2005), 237–300.

  • [12]

    Kaufmann R. M., On spineless cacti, Deligne’s conjecture and Connes–Kreimer’s Hopf algebra, Topology 46 (2007), no. 1, 39–88.

  • [13]

    Kaufmann R. M. and Schwell R., Associahedra, cyclohedra and a topological solution to the A ${A_{\infty}}$-Deligne conjecture, Adv. Math. 223 (2010), no. 6, 2166–2199.

  • [14]

    Kontsevich M., Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), no. 1, 35–72.

  • [15]

    Kontsevich M. and Soibelman Y., Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato 1999: Quantization, Deformation, and Symmetries (Dijon 1999), Math. Phys. Stud. 21, Kluwer, Dordrecht (2000), 255–307.

  • [16]

    Lambrechts P. and Volić I., Formality of the little N-disks operad, Mem. Amer. Math. Soc. 230 (2014), no. 1079, 1–116.

  • [17]

    Loday J.-L., Cyclic Homology, Grundlehren Math. Wiss. 301, Springer, Berlin, 1992.

  • [18]

    Loday J.-L. and Vallette B., Algebraic Operads, Grundlehren Math. Wiss. 346, Springer, Berlin, 2012.

  • [19]

    McClure J. E. and Smith J. H., A solution of Deligne’s Hochschild cohomology conjecture, Recent Progress in Homotopy Theory (Baltimore 2000), Contemp. Math. 293, American Mathematical Society, Providence (2002), 153–193.

  • [20]

    McClure J. E. and Smith J. H., Multivariable cochain operations and little n-cubes, J. Amer. Math. Soc. 16 (2003), no. 3, 681–704.

  • [21]

    Ševera P. and Willwacher T., Equivalence of formalities of the little discs operad, Duke Math. J. 160 (2011), no. 1, 175–206.

  • [22]

    Tamarkin D., Another proof of M. Kontsevich formality theorem, preprint 1998, https://arxiv.org/abs/math/9803025.

  • [23]

    Tamarkin D., Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1–2, 65–72.

  • [24]

    Tamarkin D., What do DG categories form?, Compos. Math. 143 (2007), no. 5, 1335–1358.

  • [25]

    Weibel C. A., An Introduction to Homological Algebra, Cambridge Stud. Adv. Math. 38, Cambridge University Press, Cambridge, 1994.

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