In this short note we compute the Chas-Sullivan BV-algebra structure on the singular homology of the free loop space of complex projective spaces. We compare this result with computations in Hochschild cohomology.
Sur l'homologie de l'espace des lacets des espaces projectifs complexes résumé : Dans cette note, on calcule l'homologie singulière de l'espace des lacets libres des espaces projectifs complexes munie de la BV-structure de Chas-Sullivan. On compare ces calculs avec ceux effectués en cohomologie de Hochschild.
Keywords: free loop spaces, Hochschild homology, string topology
Mot clés : espaces de lacets, string topologie
@article{BSMF_2011__139_4_503_0, author = {Chataur, David and Le Borgne, Jean-Fran\c{c}ois}, title = {On the loop homology of complex projective spaces}, journal = {Bulletin de la Soci\'et\'e Math\'ematique de France}, pages = {503--518}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {139}, number = {4}, year = {2011}, doi = {10.24033/bsmf.2616}, mrnumber = {2869302}, zbl = {1238.55004}, language = {en}, url = {http://archive.numdam.org/articles/10.24033/bsmf.2616/} }
TY - JOUR AU - Chataur, David AU - Le Borgne, Jean-François TI - On the loop homology of complex projective spaces JO - Bulletin de la Société Mathématique de France PY - 2011 SP - 503 EP - 518 VL - 139 IS - 4 PB - Société mathématique de France UR - http://archive.numdam.org/articles/10.24033/bsmf.2616/ DO - 10.24033/bsmf.2616 LA - en ID - BSMF_2011__139_4_503_0 ER -
%0 Journal Article %A Chataur, David %A Le Borgne, Jean-François %T On the loop homology of complex projective spaces %J Bulletin de la Société Mathématique de France %D 2011 %P 503-518 %V 139 %N 4 %I Société mathématique de France %U http://archive.numdam.org/articles/10.24033/bsmf.2616/ %R 10.24033/bsmf.2616 %G en %F BSMF_2011__139_4_503_0
Chataur, David; Le Borgne, Jean-François. On the loop homology of complex projective spaces. Bulletin de la Société Mathématique de France, Volume 139 (2011) no. 4, pp. 503-518. doi : 10.24033/bsmf.2616. http://archive.numdam.org/articles/10.24033/bsmf.2616/
[1] « Lectures on Morse theory, old and new », Bull. Amer. Math. Soc. (N.S.) 7 (1982), p. 331-358. | MR | Zbl
-[2] « Loop homology of quaternionic projective spaces », preprint arXiv:math.AT/10041550.
& -[3] « String topology », preprint arXiv:math.GT/9911159.
& -[4] « A homotopy theoretic realization of string topology », Math. Ann. 324 (2002), p. 773-798. | MR | Zbl
& -[5] « The loop homology algebra of spheres and projective spaces », in Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), Progr. Math., vol. 215, Birkhäuser, 2004, p. 77-92. | MR | Zbl
, & -[6] « Rational BV-algebra in string topology », Bull. Soc. Math. France 136 (2008), p. 311-327. | Numdam | MR | Zbl
& -[7] « The Hochschild cohomology of a closed manifold », Publ. Math. Inst. Hautes Études Sci. 99 (2004), p. 235-252. | Numdam | MR | Zbl
, & -[8] « The cohomology structure of an associative ring », Ann. of Math. 78 (1963), p. 267-288. | MR | Zbl
-[9] « Batalin-Vilkovisky algebras and two-dimensional topological field theories », Comm. Math. Phys. 159 (1994), p. 265-285. | MR | Zbl
-[10] « Loop products and closed geodesics », Duke Math. J. 150 (2009), p. 117-209. | MR | Zbl
& -[11] « String topology for complex projective spaces », preprint arXiv:math.AT/0908.1013.
-[12] « A note on the Chas-Sullivan loop product », preprint arXiv:math.GT/0903.2801.
-[13] « Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras », -Theory 32 (2004), p. 231-251. | MR | Zbl
-[14] -, « Batalin-Vilkovisky algebra structures on Hochschild cohomology », Bull. Soc. Math. France 137 (2009), p. 277-295. | Numdam | MR | Zbl
[15] -, « String topology for spheres », Comment. Math. Helv. 84 (2009), p. 135-157. | MR | Zbl
[16] « String cohomology groups of complex projective spaces », Algebr. Geom. Topol. 7 (2007), p. 2165-2238. | MR | Zbl
& -[17] « Batalin-Vilkovisky Lie algebra structure on the loop homology of complex Stiefel manifolds », Int. Math. Res. Not. 2006 (2006), Art. ID 97193, 23. | MR | Zbl
-[18] « The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products », Ann. Inst. Fourier (Grenoble) 58 (2008), p. 2351-2379. | Numdam | MR | Zbl
-[19] « A Batalin-Vilkovisky algebra structure on the Hochschild cohomology of truncated polynomials », preprint arXiv:0707.4213. | MR | Zbl
-[20] « The free loop space of globally symmetric spaces », Invent. Math. 41 (1977), p. 1-22. | MR | Zbl
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