Geodesic
The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces
Canadian journal of mathematics, Tome 71 (2019) no. 4, pp. 843-889

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-2018-021-9
Mots-clés : string topology, Batalin–Vilkovisky algebra, classifying space 
Kuribayashi, Katsuhiko; Menichi, Luc. The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces. Canadian journal of mathematics, Tome 71 (2019) no. 4, pp. 843-889. doi: 10.4153/CJM-2018-021-9
@article{10_4153_CJM_2018_021_9,
     author = {Kuribayashi, Katsuhiko and Menichi, Luc},
     title = {The {Batalin{\textendash}Vilkovisky} {Algebra} in the {String} {Topology} of {Classifying} {Spaces}},
     journal = {Canadian journal of mathematics},
     pages = {843--889},
     year = {2019},
     volume = {71},
     number = {4},
     doi = {10.4153/CJM-2018-021-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2018-021-9/}
}
TY  - JOUR
AU  - Kuribayashi, Katsuhiko
AU  - Menichi, Luc
TI  - The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces
JO  - Canadian journal of mathematics
PY  - 2019
SP  - 843
EP  - 889
VL  - 71
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2018-021-9/
DO  - 10.4153/CJM-2018-021-9
ID  - 10_4153_CJM_2018_021_9
ER  - 
%0 Journal Article
%A Kuribayashi, Katsuhiko
%A Menichi, Luc
%T The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces
%J Canadian journal of mathematics
%D 2019
%P 843-889
%V 71
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2018-021-9/
%R 10.4153/CJM-2018-021-9
%F 10_4153_CJM_2018_021_9

[1] Behrend, Kai, Ginot, Grégory, Noohi, Behrang, and Xu, Ping, String topology for stacks . Astérisque(2012), no. 343. Google Scholar

[2] Berglund, Alexander and Börjeson, Kaj, Free loop space homology of highly connected manifolds . Forum Math. 29(2017), no. 1, 201–228. . Google Scholar | DOI

[3] Bredon, Glen E., Sheaf theory . Second ed., Graduate Texts in Mathematics, 170. Springer-Verlag, New York, 1997. . Google Scholar | DOI

[4] Chas, Moira and Sullivan, Dennis, String topology. arxiv:9911159. Google Scholar

[5] Chataur, David and Le Borgne, Jean-François, On the loop homology of complex projective spaces . Bull. Soc. Math. France 139(2011), no. 4, 503–518. . Google Scholar | DOI

[6] Chataur, David and Menichi, Luc, String topology of classifying spaces . J. Reine Angew. Math. 669(2012), 1–45. . Google Scholar | DOI

[7] Earle, Clifford J. and Eells, James, Teichmüller theory for surfaces with boundary . J. Differential Geometry 4(1970), 169–185. . Google Scholar | DOI

[8] Farb, Benson and Margalit, Dan, A primer on mapping class groups . Princeton Mathematical Series, 49. Princeton University Press, Princeton, NJ, 2012. Google Scholar

[9] Félix, Yves, Halperin, Stephen, and Thomas, Jean-Claude, Rational homotopy theory . Graduate Texts in Mathematics, 205. Springer-Verlag, New York, 2001. . Google Scholar | DOI

[10] Félix, Yves, Menichi, Luc, and Thomas, Jean-Claude, Gerstenhaber duality in Hochschild cohomology . J. Pure Appl. Algebra 199(2005), no. 1–3, 43–59. . Google Scholar | DOI

[11] Félix, Yves and Thomas, Jean-Claude, Rational BV-algebra in string topology . Bull. Soc. Math. France 136(2008), no. 2, 311–327. . Google Scholar | DOI

[12] Félix, Yves and Thomas, Jean-Claude, String topology on Gorenstein spaces . Math. Ann. 345(2009), no. 2, 417–452. . Google Scholar | DOI

[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, 947–1007. . Google Scholar | DOI

[14] Godin, Véronique, Higher string topology operations. arxiv:0711.4859. Google Scholar

[15] Grodal, Jesper and Lahtinen, Anssi, String topology of finite groups of lie type. http://www.math.ku.dk/∼jg/papers/stringtoplie.pdf, July2017. Google Scholar

[16] Halperin, Stephen, Universal enveloping algebras and loop space homology . J. Pure Appl. Algebra 83(1992), no. 3, 237–282. . Google Scholar | DOI

[17] Hamstrom, Mary-Elizabeth, Homotopy groups of the space of homeomorphisms on a 2-manifold . Illinois J. Math. 10(1966), 563–573. Google Scholar

[18] Hepworth, Richard, String topology for complex projective spaces. 2009. arxiv:0908.1013. Google Scholar

[19] Hepworth, Richard, String topology for Lie groups . J. Topol. 3(2010), no. 2, 424–442. . Google Scholar | DOI

[20] Hepworth, Richard and Lahtinen, Anssi, On string topology of classifying spaces . Adv. Math. 281(2015), 394–507. . Google Scholar | DOI

[21] Iwase, Norio, Adjoint action of a finite loop space . Proc. Amer. Math. Soc. 125(1997), no. 9, 2753–2757. . Google Scholar | DOI

[22] Johnson, Dennis L., Homeomorphisms of a surface which act trivially on homology . Proc. Amer. Math. Soc. 75(1979), no. 1, 119–125. . Google Scholar | DOI

[23] Keller, Bernhard, Deformation quantization after Kontsevich and Tamarkin . In: Déformation, quantification, théorie de Lie . Panor. Synthèses, 20. Soc. Math. France, Paris, 2005, pp. 19–62. Google Scholar

[24] Kishimoto, Daisuke and Kono, Akira, On the cohomology of free and twisted loop spaces . J. Pure Appl. Algebra 214(2010), no. 5, 646–653. . Google Scholar | DOI

[25] Kock, Joachim, Frobenius algebras and 2D topological quantum field theories . London Mathematical Society Student Texts, 59. Cambridge University Press, Cambridge, 2004. Google Scholar

[26] Kono, Akira and Kuribayashi, Katsuhiko, Module derivations and cohomological splitting of adjoint bundles . Fund. Math. 180(2003), no. 3, 199–221. . Google Scholar | DOI

[27] Kupers, Alexander, String topology operations. Master’s thesis, Utrecht University, The Netherlands, 2011. Google Scholar

[28] Kuribayashi, Katsuhiko, Module derivations and the adjoint action of a finite loop space . J. Math. Kyoto Univ. 39(1999), no. 1, 67–85. . Google Scholar | DOI

[29] Kuribayashi, Katsuhiko, Menichi, Luc, and Naito, Takahito, Derived string topology and the Eilenberg-Moore spectral sequence . Israel J. Math. 209(2015), no. 2, 745–802. . Google Scholar | DOI

[30] Lahtinen, Anssi, Higher operations in string topology of classifying spaces . Math. Ann. 368(2017), no. 1-2, 1–63. . Google Scholar | DOI

[31] Mccleary, John, A user’s guide to spectral sequences Second ed., Cambridge Studies in Advanced Mathematics, 58, Cambridge University Press, Cambridge, 2001. Google Scholar

[32] Menichi, Luc, The cohomology ring of free loop spaces . Homology Homotopy Appl. 3(2001), no. 1, 193–224. . Google Scholar | DOI

[33] Menichi, Luc, On the cohomology algebra of a fiber . Algebr. Geom. Topol. 1(2001), 719–742. . Google Scholar | DOI

[34] Menichi, Luc, String topology for spheres . Comment. Math. Helv. 84(2009), no. 1, 135–157. . Google Scholar | DOI

[35] Menichi, Luc, A Batalin-Vilkovisky algebra morphism from double loop spaces to free loops . Trans. Amer. Math. Soc. 363(2011), no. 8, 4443–4462. . Google Scholar | DOI

[36] Milnor, John W. and Moore, John C., On the structure of Hopf algebras . Ann. of Math. (2) 81(1965), 211–264. . Google Scholar | DOI

[37] Milnor, John W. and Stasheff, James D., Characteristic classes . Annals of Mathematics Studies, 76, Princeton University Press, Princeton, NJ, 1974. Google Scholar

[38] Mimura, Mamoru and Toda, Hirosi, Topology of Lie groups. I, II . Translations of Mathematical Monographs, 91. American Mathematical Society, Providence, RI, 1991. Google Scholar

[39] Spanier, Edwin H., Algebraic topology . Springer-Verlag, New York, 1981. Google Scholar

[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. 567–577. Google Scholar

[41] Tamanoi, Hirotaka, Batalin-Vilkovisky Lie algebra structure on the loop homology of complex Stiefel manifolds . Int. Math. Res. Not. (2006), 1–23. . Google Scholar | DOI

[42] Tamanoi, Hirotaka, Cap products in string topology . Algebr. Geom. Topol. 9(2009), no. 2, 1201–1224. . Google Scholar | DOI

[43] Tamanoi, Hirotaka, Stable string operations are trivial . Int. Math. Res. Not. IMRN (2009), no. 24, 4642–4685. . Google Scholar | DOI

[44] Tamanoi, Hirotaka, Loop coproducts in string topology and triviality of higher genus TQFT operations . J. Pure Appl. Algebra 214(2010), no. 5, 605–615. . Google Scholar | DOI

[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. Google Scholar

[46] Wahl, Nathalie, Ribbon braids and related operads. Ph.D. thesis, Oxford University, 2001. http://www.math.ku.dk/∼wahl/. Google Scholar

[47] Westerland, Craig, String homology of spheres and projective spaces . Algebr. Geom. Topol. 7(2007), 309–325. . Google Scholar | DOI

[48] Yang, Tian, A Batalin-Vilkovisky algebra structure on the Hochschild cohomology of truncated polynomials . Topology Appl. 160(2013), no. 13, 1633–1651. . Google Scholar | DOI

Cité par Sources :