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.
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.
@article{BSMF_2011__139_4_503_0,
author = {Chataur, David and Le Borgne, Jean-Fran\c cois},
title = {On the loop homology of complex projective spaces},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
volume = {139},
year = {2011},
pages = {503-518},
doi = {10.24033/bsmf.2616},
mrnumber = {2869302},
zbl = {1238.55004},
language = {en},
url = {http://dml.mathdoc.fr/item/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, Tome 139 (2011) pp. 503-518. doi : 10.24033/bsmf.2616. http://gdmltest.u-ga.fr/item/BSMF_2011__139_4_503_0/
[1] - « Lectures on Morse theory, old and new », Bull. Amer. Math. Soc. (N.S.) 7 (1982), p. 331-358. | MR 663786 | Zbl 0505.58001
[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 1942249 | Zbl 1025.55005
[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 2039760 | Zbl 1054.55006
[6] & - « Rational BV-algebra in string topology », Bull. Soc. Math. France 136 (2008), p. 311-327. | Numdam | MR 2415345 | Zbl 1160.55006
[7] , & - « The Hochschild cohomology of a closed manifold », Publ. Math. Inst. Hautes Études Sci. 99 (2004), p. 235-252. | Numdam | MR 2075886 | Zbl 1060.57019
[8] - « The cohomology structure of an associative ring », Ann. of Math. 78 (1963), p. 267-288. | MR 161898 | Zbl 0131.27302
[9] - « Batalin-Vilkovisky algebras and two-dimensional topological field theories », Comm. Math. Phys. 159 (1994), p. 265-285. | MR 1256989 | Zbl 0807.17026
[10] & - « Loop products and closed geodesics », Duke Math. J. 150 (2009), p. 117-209. | MR 2560110 | Zbl 1181.53036
[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 2114167 | Zbl 1101.19003
[14] -, « Batalin-Vilkovisky algebra structures on Hochschild cohomology », Bull. Soc. Math. France 137 (2009), p. 277-295. | Numdam | MR 2543477 | Zbl 1180.16007
[15] -, « String topology for spheres », Comment. Math. Helv. 84 (2009), p. 135-157. | MR 2466078 | Zbl 1159.55004
[16] & - « String cohomology groups of complex projective spaces », Algebr. Geom. Topol. 7 (2007), p. 2165-2238. | MR 2366191 | Zbl 1134.55005
[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 2211159 | Zbl 1108.55007
[18] - « The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products », Ann. Inst. Fourier (Grenoble) 58 (2008), p. 2351-2379. | Numdam | MR 2498354 | Zbl 1218.16004
[19] - « A Batalin-Vilkovisky algebra structure on the Hochschild cohomology of truncated polynomials », preprint arXiv:0707.4213. | MR 3091339 | Zbl 1282.55013
[20] - « The free loop space of globally symmetric spaces », Invent. Math. 41 (1977), p. 1-22. | MR 649625 | Zbl 0338.58007