Swiss-cheese action on the totalization of action-operads

Ducoulombier, Julien¹

¹LAGA, CNRS, UMR 7539, Université Paris 13, 99 avenue Jean-Batiste Clément, 93430 Villetaneuse, France

Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1683-1726

Résumé

We prove that, if a pair of semicosimplicial spaces (Xc∙;Xo∙) arises from a colored operad, then the semitotalization  sTot(Xo∙) has the homotopy type of a relative double loop space and the pair ( sTot(Xc∙); sTot(Xo∙)) is weakly equivalent to an explicit algebra over the two dimensional Swiss-cheese operad SC2.

Détail

DOI : 10.2140/agt.2016.16.1683

Classification : 55P48, 55P47
Keywords: colored operads, loop spaces, cosimplicial spaces, model categories

Bibliographie

[1] G Arone, V Turchin, On the rational homology of high-dimensional analogues of spaces of long knots, Geom. Topol. 18 (2014) 1261

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

[3] C Berger, I Moerdijk, The Boardman–Vogt resolution of operads in monoidal model categories, Topology 45 (2006) 807

[4] C Berger, I Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, from: "Categories in algebra, geometry and mathematical physics" (editors A Davydov, M Batanin, M Johnson, S Lack, A Neeman), Contemp. Math. 431, Amer. Math. Soc. (2007) 31

[5] J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, 347, Springer (1973)

[6] E Dror, W G Dwyer, A long homology localization tower, Comment. Math. Helv. 52 (1977) 185

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

[8] W G Dwyer, J Spaliński, Homotopy theories and model categories, from: "Handbook of algebraic topology" (editor I M James), North-Holland (1995) 73

[9] B Fresse, Modules over operads and functors, 1967, Springer (2009)

[10] J E Harper, Homotopy theory of modules over operads and non–Σ operads in monoidal model categories, J. Pure Appl. Algebra 214 (2010) 1407

[11] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[12] P S Hirschhorn, Model categories and their localizations, 99, Amer. Math. Soc. (2003)

[13] E Hoefel, M Livernet, J Stasheff, A∞–actions and recognition of relative loop spaces, preprint (2013)

[14] M Hovey, Model categories, 63, Amer. Math. Soc. (1999)

[15] 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 (2004) 133

[16] M Robertson, Spaces of operad structures, preprint (2011)

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

[18] S Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957) 604

[19] V Turchin, Delooping totalization of a multiplicative operad, J. Homotopy Relat. Struct. 9 (2014) 349

[20] R M Vogt, Cofibrant operads and universal E∞ operads, Topology Appl. 133 (2003) 69

[21] A A Voronov, The Swiss-cheese operad, from: "Homotopy invariant algebraic structures" (editors J P Meyer, J Morava, W S Wilson), Contemp. Math. 239, Amer. Math. Soc. (1999) 365