Skip to main content Accessibility help
×
Home

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

  • Katsuhiko Kuribayashi (a1) and Luc Menichi (a2)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

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

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Copyright

Footnotes

Hide All

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

Footnotes

References

Hide All
[1] Behrend, Kai, Ginot, Grégory, Noohi, Behrang, and Xu, Ping, String topology for stacks . Astérisque(2012), no. 343.
[2] Berglund, Alexander and Börjeson, Kaj, Free loop space homology of highly connected manifolds . Forum Math. 29(2017), no. 1, 201228. https://doi.org/10.1515/forum-2015-0074.
[3] Bredon, Glen E., Sheaf theory . Second ed., Graduate Texts in Mathematics, 170. Springer-Verlag, New York, 1997. https://doi.org/10.1007/978-1-4612-0647-7.
[4] Chas, Moira and Sullivan, Dennis, String topology. arxiv:9911159.
[5] Chataur, David and Le Borgne, Jean-François, On the loop homology of complex projective spaces . Bull. Soc. Math. France 139(2011), no. 4, 503518. https://doi.org/10.24033/bsmf.2616.
[6] Chataur, David and Menichi, Luc, String topology of classifying spaces . J. Reine Angew. Math. 669(2012), 145. https://doi.org/10.1515/CRELLE.2011.140.
[7] Earle, Clifford J. and Eells, James, Teichmüller theory for surfaces with boundary . J. Differential Geometry 4(1970), 169185. https://doi.org/10.4310/jdg/1214429381.
[8] Farb, Benson and Margalit, Dan, A primer on mapping class groups . Princeton Mathematical Series, 49. Princeton University Press, Princeton, NJ, 2012.
[9] Félix, Yves, Halperin, Stephen, and Thomas, Jean-Claude, Rational homotopy theory . Graduate Texts in Mathematics, 205. Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4613-0105-9.
[10] Félix, Yves, Menichi, Luc, and Thomas, Jean-Claude, Gerstenhaber duality in Hochschild cohomology . J. Pure Appl. Algebra 199(2005), no. 1–3, 4359. https://doi.org/10.1016/j.jpaa.2004.11.004.
[11] Félix, Yves and Thomas, Jean-Claude, Rational BV-algebra in string topology . Bull. Soc. Math. France 136(2008), no. 2, 311327. https://doi.org/10.24033/bsmf.2558.
[12] Félix, Yves and Thomas, Jean-Claude, String topology on Gorenstein spaces . Math. Ann. 345(2009), no. 2, 417452. https://doi.org/10.1007/s00208-009-0361-5.
[13] 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. https://doi.org/10.4007/annals.2011.174.2.5.
[14] Godin, Véronique, Higher string topology operations. arxiv:0711.4859.
[15] Grodal, Jesper and Lahtinen, Anssi, String topology of finite groups of lie type. http://www.math.ku.dk/∼jg/papers/stringtoplie.pdf, July2017.
[16] Halperin, Stephen, Universal enveloping algebras and loop space homology . J. Pure Appl. Algebra 83(1992), no. 3, 237282. https://doi.org/10.1016/0022-4049(92)90046-I.
[17] Hamstrom, Mary-Elizabeth, Homotopy groups of the space of homeomorphisms on a 2-manifold . Illinois J. Math. 10(1966), 563573.
[18] Hepworth, Richard, String topology for complex projective spaces. 2009. arxiv:0908.1013.
[19] Hepworth, Richard, String topology for Lie groups . J. Topol. 3(2010), no. 2, 424442. https://doi.org/10.1112/jtopol/jtq012.
[20] Hepworth, Richard and Lahtinen, Anssi, On string topology of classifying spaces . Adv. Math. 281(2015), 394507. https://doi.org/10.1016/j.aim.2015.03.022.
[21] Iwase, Norio, Adjoint action of a finite loop space . Proc. Amer. Math. Soc. 125(1997), no. 9, 27532757. https://doi.org/10.1090/S0002-9939-97-03924-5.
[22] Johnson, Dennis L., Homeomorphisms of a surface which act trivially on homology . Proc. Amer. Math. Soc. 75(1979), no. 1, 119125. https://doi.org/10.2307/2042686.
[23] Keller, Bernhard, Deformation quantization after Kontsevich and Tamarkin . In: Déformation, quantification, théorie de Lie . Panor. Synthèses, 20. Soc. Math. France, Paris, 2005, pp. 1962.
[24] Kishimoto, Daisuke and Kono, Akira, On the cohomology of free and twisted loop spaces . J. Pure Appl. Algebra 214(2010), no. 5, 646653. https://doi.org/10.1016/j.jpaa.2009.07.006.
[25] Kock, Joachim, Frobenius algebras and 2D topological quantum field theories . London Mathematical Society Student Texts, 59. Cambridge University Press, Cambridge, 2004.
[26] Kono, Akira and Kuribayashi, Katsuhiko, Module derivations and cohomological splitting of adjoint bundles . Fund. Math. 180(2003), no. 3, 199221. https://doi.org/10.4064/fm180-3-1.
[27] Kupers, Alexander, String topology operations. Master’s thesis, Utrecht University, The Netherlands, 2011.
[28] Kuribayashi, Katsuhiko, Module derivations and the adjoint action of a finite loop space . J. Math. Kyoto Univ. 39(1999), no. 1, 6785. https://doi.org/10.1215/kjm/1250517954.
[29] Kuribayashi, Katsuhiko, Menichi, Luc, and Naito, Takahito, Derived string topology and the Eilenberg-Moore spectral sequence . Israel J. Math. 209(2015), no. 2, 745802. https://doi.org/10.1007/s11856-015-1236-y.
[30] Lahtinen, Anssi, Higher operations in string topology of classifying spaces . Math. Ann. 368(2017), no. 1-2, 163. https://doi.org/10.1007/s00208-016-1406-1.
[31] McCleary, John, A user’s guide to spectral sequences Second ed., Cambridge Studies in Advanced Mathematics, 58, Cambridge University Press, Cambridge, 2001.
[32] Menichi, Luc, The cohomology ring of free loop spaces . Homology Homotopy Appl. 3(2001), no. 1, 193224. https://doi.org/10.4310/HHA.2001.v3.n1.a9.
[33] Menichi, Luc, On the cohomology algebra of a fiber . Algebr. Geom. Topol. 1(2001), 719742. https://doi.org/10.2140/agt.2001.1.719.
[34] Menichi, Luc, String topology for spheres . Comment. Math. Helv. 84(2009), no. 1, 135157. https://doi.org/10.4171/CMH/155.
[35] Menichi, Luc, A Batalin-Vilkovisky algebra morphism from double loop spaces to free loops . Trans. Amer. Math. Soc. 363(2011), no. 8, 44434462. https://doi.org/10.1090/S0002-9947-2011-05374-2.
[36] Milnor, John W. and Moore, John C., On the structure of Hopf algebras . Ann. of Math. (2) 81(1965), 211264. https://doi.org/10.2307/1970615.
[37] Milnor, John W. and Stasheff, James D., Characteristic classes . Annals of Mathematics Studies, 76, Princeton University Press, Princeton, NJ, 1974.
[38] Mimura, Mamoru and Toda, Hirosi, Topology of Lie groups. I, II . Translations of Mathematical Monographs, 91. American Mathematical Society, Providence, RI, 1991.
[39] Spanier, Edwin H., Algebraic topology . Springer-Verlag, New York, 1981.
[40] 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.
[41] Tamanoi, Hirotaka, Batalin-Vilkovisky Lie algebra structure on the loop homology of complex Stiefel manifolds . Int. Math. Res. Not. (2006), 123. https://doi.org/10.1155/IMRN/2006/97193.
[42] Tamanoi, Hirotaka, Cap products in string topology . Algebr. Geom. Topol. 9(2009), no. 2, 12011224. https://doi.org/10.2140/agt.2009.9.1201.
[43] Tamanoi, Hirotaka, Stable string operations are trivial . Int. Math. Res. Not. IMRN (2009), no. 24, 46424685. https://doi.org/10.1093/imrn/rnp104.
[44] Tamanoi, Hirotaka, Loop coproducts in string topology and triviality of higher genus TQFT operations . J. Pure Appl. Algebra 214(2010), no. 5, 605615. https://doi.org/10.1016/j.jpaa.2009.07.011.
[45] Tezuka, Michishige, On the cohomology of finite chevalley groups and free loop spaces of classifying spaces. Suurikenkoukyuuroku, 1057:54–55, 1998. http://hdl.handle.net/2433/62316.
[46] Wahl, Nathalie, Ribbon braids and related operads. Ph.D. thesis, Oxford University, 2001. http://www.math.ku.dk/∼wahl/.
[47] Westerland, Craig, String homology of spheres and projective spaces . Algebr. Geom. Topol. 7(2007), 309325. https://doi.org/10.2140/agt.2007.7.309.
[48] Yang, Tian, A Batalin-Vilkovisky algebra structure on the Hochschild cohomology of truncated polynomials . Topology Appl. 160(2013), no. 13, 16331651. https://doi.org/10.1016/j.topol.2013.06.010.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed