Formality of Sinha’s cosimplicial model for long knots spaces and the Gerstenhaber algebra structure of homology

Songhafouo Tsopméné, Paul Arnaud¹

¹Université catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

Algebraic and Geometric Topology, Tome 13 (2013) no. 4, pp. 2193-2205

Résumé

Sinha constructed a cosimplicial space KN∙ that gives a model for the space of long knots modulo immersions in ℝN, N ≥ 4. On the other hand, Lambrechts, Turchin and Volić showed that for N ≥ 4 the homology Bousfield–Kan spectral sequence associated to Sinha’s cosimplicial space KN∙ collapses at the E2 page rationally. Their approach consists in first proving the formality of some other diagrams approximating KN∙ and next deducing the collapsing result. In this paper, we prove directly the formality of Sinha’s cosimplicial space, which immediately implies the collapsing result for N ≥ 3. Moreover, we prove that the isomorphism between the E2 page and the homology of the space of long knots modulo immersions respects the Gerstenhaber algebra structure, when N ≥ 4.

Détail

DOI : 10.2140/agt.2013.13.2193

Classification : 57Q45, 57Q45, 18D50, 55P48, 17B63
Keywords: multiplicative operads, model categories, long knots

Bibliographie

[1] C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805

[2] R Budney, Little cubes and long knots, Topology 46 (2007) 1

[3] W Dwyer, K Hess, Long knots and maps between operads, Geom. Topol. 16 (2012) 919

[4] B Fresse, Modules over operads and functors, Lecture Notes in Mathematics 1967, Springer (2009)

[5] M Gerstenhaber, A A Voronov, Homotopy $G$–algebras and moduli space operad, Internat. Math. Res. Notices (1995) 141

[6] M Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society (1999)

[7] M Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999) 35

[8] P Lambrechts, V Turchin, I Volić, The rational homology of spaces of long knots in codimension $\gt 2$, Geom. Topol. 14 (2010) 2151

[9] P Lambrechts, I Volić, Formality of the little $N$–disks operad (2012)

[10] J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics, Springer (1972)

[11] J E Mcclure, J H Smith, Cosimplicial objects and little $n$–cubes, I, Amer. J. Math. 126 (2004) 1109

[12] J E Mcclure, J H Smith, Operads and cosimplicial objects: an introduction, from: "Axiomatic, enriched and motivic homotopy theory" (editor J P C Greenlees), NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer Acad. Publ. (2004) 133

[13] S Moriya, Sinha's spectral sequence and homotopical algebra of operads

[14] F Muro, Homotopy theory of nonsymmetric operads, Algebr. Geom. Topol. 11 (2011) 1541

[15] K Sakai, Poisson structures on the homology of the space of knots, from: "Groups, homotopy and configuration spaces" (editors N Iwase, T Kohno, R Levi, D Tamaki, J Wu), Geom. Topol. Monogr. 13 (2008) 463

[16] P Salvatore, Knots, operads, and double loop spaces, Int. Math. Res. Not. 2006 (2006) 22

[17] D P Sinha, Operads and knot spaces, J. Amer. Math. Soc. 19 (2006) 461

[18] M Spitzweck, Operads, Algebras and Modules in General Model Categories

[19] V Turchin, Delooping totalization of a multiplicative operad