Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-18T04:30:14.989Z Has data issue: false hasContentIssue false

The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces

Published online by Cambridge University Press:  09 January 2019

Katsuhiko Kuribayashi
Department of Mathematical Sciences, Faculty of Science, Shinshu University, Nagano 390-8621, Japan Email:
Luc Menichi
LAREMA - UMR CNRS 6093, Université d’Angers, 2 Boulevard Lavoisier, 49045 Angers, France Email:
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For almost any compact connected Lie group $G$ and any field $\mathbb{F}_{p}$, we compute the Batalin–Vilkovisky algebra $H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$ on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if $p$ is odd or $p=0$, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology $HH^{\star }(H_{\star }(G),H_{\star }(G))$. Over $\mathbb{F}_{2}$ , such an isomorphism of Batalin–Vilkovisky algebras does not hold when $G=\text{SO}(3)$ or $G=G_{2}$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.

© Canadian Mathematical Society 2018 


The first author was partially supported by JSPS KAKENHI Grant Number 25287008.


Behrend, Kai, Ginot, Grégory, Noohi, Behrang, and Xu, Ping, String topology for stacks . Astérisque(2012), no. 343.Google Scholar
Berglund, Alexander and Börjeson, Kaj, Free loop space homology of highly connected manifolds . Forum Math. 29(2017), no. 1, 201228. Scholar
Bredon, Glen E., Sheaf theory . Second ed., Graduate Texts in Mathematics, 170. Springer-Verlag, New York, 1997. Scholar
Chas, Moira and Sullivan, Dennis, String topology. arxiv:9911159.Google Scholar
Chataur, David and Le Borgne, Jean-François, On the loop homology of complex projective spaces . Bull. Soc. Math. France 139(2011), no. 4, 503518. Scholar
Chataur, David and Menichi, Luc, String topology of classifying spaces . J. Reine Angew. Math. 669(2012), 145. Scholar
Earle, Clifford J. and Eells, James, Teichmüller theory for surfaces with boundary . J. Differential Geometry 4(1970), 169185. Scholar
Farb, Benson and Margalit, Dan, A primer on mapping class groups . Princeton Mathematical Series, 49. Princeton University Press, Princeton, NJ, 2012.Google Scholar
Félix, Yves, Halperin, Stephen, and Thomas, Jean-Claude, Rational homotopy theory . Graduate Texts in Mathematics, 205. Springer-Verlag, New York, 2001. Scholar
Félix, Yves, Menichi, Luc, and Thomas, Jean-Claude, Gerstenhaber duality in Hochschild cohomology . J. Pure Appl. Algebra 199(2005), no. 1–3, 4359. Scholar
Félix, Yves and Thomas, Jean-Claude, Rational BV-algebra in string topology . Bull. Soc. Math. France 136(2008), no. 2, 311327. Scholar
Félix, Yves and Thomas, Jean-Claude, String topology on Gorenstein spaces . Math. Ann. 345(2009), no. 2, 417452. Scholar
Freed, Daniel S., Hopkins, Michael J., and Teleman, Constantin, Loop groups and twisted K-theory III . Ann. of Math. (2) 174(2011), no. 2, 9471007. Scholar
Godin, Véronique, Higher string topology operations. arxiv:0711.4859.Google Scholar
Grodal, Jesper and Lahtinen, Anssi, String topology of finite groups of lie type.∼jg/papers/stringtoplie.pdf, July2017.Google Scholar
Halperin, Stephen, Universal enveloping algebras and loop space homology . J. Pure Appl. Algebra 83(1992), no. 3, 237282. Scholar
Hamstrom, Mary-Elizabeth, Homotopy groups of the space of homeomorphisms on a 2-manifold . Illinois J. Math. 10(1966), 563573.Google Scholar
Hepworth, Richard, String topology for complex projective spaces. 2009. arxiv:0908.1013.Google Scholar
Hepworth, Richard, String topology for Lie groups . J. Topol. 3(2010), no. 2, 424442. Scholar
Hepworth, Richard and Lahtinen, Anssi, On string topology of classifying spaces . Adv. Math. 281(2015), 394507. Scholar
Iwase, Norio, Adjoint action of a finite loop space . Proc. Amer. Math. Soc. 125(1997), no. 9, 27532757. Scholar
Johnson, Dennis L., Homeomorphisms of a surface which act trivially on homology . Proc. Amer. Math. Soc. 75(1979), no. 1, 119125. Scholar
Keller, Bernhard, Deformation quantization after Kontsevich and Tamarkin . In: Déformation, quantification, théorie de Lie . Panor. Synthèses, 20. Soc. Math. France, Paris, 2005, pp. 1962.Google Scholar
Kishimoto, Daisuke and Kono, Akira, On the cohomology of free and twisted loop spaces . J. Pure Appl. Algebra 214(2010), no. 5, 646653. Scholar
Kock, Joachim, Frobenius algebras and 2D topological quantum field theories . London Mathematical Society Student Texts, 59. Cambridge University Press, Cambridge, 2004.Google Scholar
Kono, Akira and Kuribayashi, Katsuhiko, Module derivations and cohomological splitting of adjoint bundles . Fund. Math. 180(2003), no. 3, 199221. Scholar
Kupers, Alexander, String topology operations. Master’s thesis, Utrecht University, The Netherlands, 2011.Google Scholar
Kuribayashi, Katsuhiko, Module derivations and the adjoint action of a finite loop space . J. Math. Kyoto Univ. 39(1999), no. 1, 6785. Scholar
Kuribayashi, Katsuhiko, Menichi, Luc, and Naito, Takahito, Derived string topology and the Eilenberg-Moore spectral sequence . Israel J. Math. 209(2015), no. 2, 745802. Scholar
Lahtinen, Anssi, Higher operations in string topology of classifying spaces . Math. Ann. 368(2017), no. 1-2, 163. Scholar
McCleary, John, A user’s guide to spectral sequences Second ed., Cambridge Studies in Advanced Mathematics, 58, Cambridge University Press, Cambridge, 2001.Google Scholar
Menichi, Luc, The cohomology ring of free loop spaces . Homology Homotopy Appl. 3(2001), no. 1, 193224. Scholar
Menichi, Luc, On the cohomology algebra of a fiber . Algebr. Geom. Topol. 1(2001), 719742. Scholar
Menichi, Luc, String topology for spheres . Comment. Math. Helv. 84(2009), no. 1, 135157. Scholar
Menichi, Luc, A Batalin-Vilkovisky algebra morphism from double loop spaces to free loops . Trans. Amer. Math. Soc. 363(2011), no. 8, 44434462. Scholar
Milnor, John W. and Moore, John C., On the structure of Hopf algebras . Ann. of Math. (2) 81(1965), 211264. Scholar
Milnor, John W. and Stasheff, James D., Characteristic classes . Annals of Mathematics Studies, 76, Princeton University Press, Princeton, NJ, 1974.Google Scholar
Mimura, Mamoru and Toda, Hirosi, Topology of Lie groups. I, II . Translations of Mathematical Monographs, 91. American Mathematical Society, Providence, RI, 1991.Google Scholar
Spanier, Edwin H., Algebraic topology . Springer-Verlag, New York, 1981.Google Scholar
Stasheff, James and Halperin, Steve, Differential algebra in its own rite . In: Proceedings of the Advanced Study Institute on Algebraic Topology, vol. 3 . Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 567577.Google Scholar
Tamanoi, Hirotaka, Batalin-Vilkovisky Lie algebra structure on the loop homology of complex Stiefel manifolds . Int. Math. Res. Not. (2006), 123. Scholar
Tamanoi, Hirotaka, Cap products in string topology . Algebr. Geom. Topol. 9(2009), no. 2, 12011224. Scholar
Tamanoi, Hirotaka, Stable string operations are trivial . Int. Math. Res. Not. IMRN (2009), no. 24, 46424685. Scholar
Tamanoi, Hirotaka, Loop coproducts in string topology and triviality of higher genus TQFT operations . J. Pure Appl. Algebra 214(2010), no. 5, 605615. Scholar
Tezuka, Michishige, On the cohomology of finite chevalley groups and free loop spaces of classifying spaces. Suurikenkoukyuuroku, 1057:54–55, 1998. Scholar
Wahl, Nathalie, Ribbon braids and related operads. Ph.D. thesis, Oxford University, 2001.∼wahl/.Google Scholar
Westerland, Craig, String homology of spheres and projective spaces . Algebr. Geom. Topol. 7(2007), 309325. Scholar
Yang, Tian, A Batalin-Vilkovisky algebra structure on the Hochschild cohomology of truncated polynomials . Topology Appl. 160(2013), no. 13, 16331651. Scholar