Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

6 Issues per year

IMPACT FACTOR 2017: 0.695
5-year IMPACT FACTOR: 0.750

CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 0.966
Source Normalized Impact per Paper (SNIP) 2017: 0.889

Mathematical Citation Quotient (MCQ) 2016: 0.75

See all formats and pricing
More options …
Volume 29, Issue 2


A direct computation of the cohomology of the braces operad

Vasily Dolgushev
  • Corresponding author
  • Department of Mathematics, Temple University, Wachman Hall , Rm. 638, 1805 N. Broad St., Philadelphia, PA, 19122, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Thomas Willwacher
Published Online: 2016-06-30 | DOI: https://doi.org/10.1515/forum-2016-0123


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.

Keywords: Algebraic operads; homotopy algebras; the Deligne conjecture on Hochschild cochains

MSC 2010: 18D50; 18G55; 55P10


  • [1]

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

  • [2]

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

  • [3]

    Deligne P., Letter to V. Drinfeld, M. Gerstenhaber, J. P. May, V. Schechtman and J. Stasheff, unpublished, 1993.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

  • [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. Google Scholar

  • [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. Google Scholar

  • [6]

    Drinfeld V. G., On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(¯/) (in Russian), Algebra i Analiz 2 (1990), no. 4, 149–181. Google Scholar

  • [7]

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

  • [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. Google Scholar

  • [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. Google Scholar

  • [11]

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

  • [12]

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

  • [13]

    Kaufmann R. M. and Schwell R., Associahedra, cyclohedra and a topological solution to the A-Deligne conjecture, Adv. Math. 223 (2010), no. 6, 2166–2199. Google Scholar

  • [14]

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

  • [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. Google Scholar

  • [16]

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

  • [17]

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

  • [18]

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

  • [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. Google Scholar

  • [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. Google Scholar

  • [21]

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

  • [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. Google Scholar

  • [24]

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

  • [25]

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

About the article

Received: 2016-05-28

Published Online: 2016-06-30

Published in Print: 2017-03-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1161867

Award identifier / Grant number: DMS-1501001

Funding Source: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

Award identifier / Grant number: 200021_150012

Award identifier / Grant number: SwissMAP NCCR

Vasily Dolgushev has been supported partially by NSF grants DMS-1161867 and DMS-1501001 and Thomas Willwacher has been supported partially by the Swiss National Science foundation, grant 200021_150012, and the SwissMAP NCCR funded by the Swiss National Science foundation.

Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 465–488, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0123.

Export Citation

© 2017 by De Gruyter.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Boris Shoikhet
International Mathematics Research Notices, 2018, Volume 2018, Number 19, Page 5857
V.A. Dolgushev and G.E. Schneider
Journal of Pure and Applied Algebra, 2018

Comments (0)

Please log in or register to comment.
Log in