On the rational homology of high-dimensional analogues of spaces of long knots

Arone, Gregory¹ ; Turchin, Victor²

¹ Department of Mathematics, University of Virginia, Kerchof Hall, PO Box 400137, Charlottesville, VA 22904, USA
² Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, KS 66506, USA

Geometry & topology, Tome 18 (2014) no. 3, pp. 1261-1322.

Résumé

We study high-dimensional analogues of spaces of long knots. These are spaces of compactly supported embeddings (modulo immersions) of ${ℝ}^m$ into ${ℝ}^n$. We view the space of embeddings as the value of a certain functor at ${ℝ}^m$, and we apply manifold calculus to this functor. Our first result says that the Taylor tower of this functor can be expressed as the space of maps between infinitesimal bimodules over the little-disks operad. We then show that the formality of the little-disks operad has implications for the homological behavior of the Taylor tower. Our second result says that when $2m+1<n$, the singular chain complex of these spaces of embeddings is rationally equivalent to a direct sum of certain finite chain complexes, which we describe rather explicitly.

Détail

DOI : 10.2140/gt.2014.18.1261

Classification : 57R70, 18D50, 18G55
Keywords: embedding spaces, infinitesimal bimodules, formality

Bibliographie

[1] S T Ahearn, N J Kuhn, Product and other fine structure in polynomial resolutions of mapping spaces, Algebr. Geom. Topol. 2 (2002) 591 | DOI

[2] V I Arnold, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969) 227

[3] G Arone, A note on the homology of Σn, the Schwartz genus, and solving polynomial equations, from: "An alpine anthology of homotopy theory" (editors D Arlettaz, K Hess), Contemp. Math. 399, Amer. Math. Soc. (2006) 1 | DOI

[4] G Arone, Derivatives of embedding functors, I : The stable case, J. Topol. 2 (2009) 461 | DOI

[5] G Arone, P Lambrechts, I Volić, Calculus of functors, operad formality, and rational homology of embedding spaces, Acta Math. 199 (2007) 153 | DOI

[6] P Boavida De Brito, M S Weiss, Manifold calculus and homotopy sheaves, Homology Homotopy Appl. 15 (2013) 361

[7] A S Cattaneo, P Cotta-Ramusino, R Longoni, Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol. 2 (2002) 949 | DOI

[8] M Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005) 833 | DOI

[9] F R Cohen, T J Lada, J P May, The homology of iterated loop spaces, 533, Springer (1976)

[10] F R Cohen, L R Taylor, On the representation theory associated to the cohomology of configuration spaces, from: "Algebraic topology" (editor M C Tangora), Contemp. Math. 146, Amer. Math. Soc. (1993) 91 | DOI

[11] D Dugger, D C Isaksen, Topological hypercovers and A1–realizations, Math. Z. 246 (2004) 667 | DOI

[12] S Eilenberg, S Mac Lane, On the groups of H(Π,n), I, Ann. of Math. 58 (1953) 55 | DOI

[13] B Fresse, Koszul duality of operads and homology of partition posets, from: "Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic –theory" (editors P Goerss, S Priddy), Contemp. Math. 346, Amer. Math. Soc. (2004) 115 | DOI

[14] M W Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959) 242 | DOI

[15] M Kontsevich, Y Soibelman, Deformations of algebras over operads and the Deligne conjecture, from: "Conférence Moshé Flato 1999, Vol. I" (editors G Dito, D Sternheimer), Math. Phys. Stud. 21, Kluwer Acad. Publ. (2000) 255

[16] P Lambrechts, V Turchin, I Volić, The rational homology of spaces of long knots in codimension greater than 2, Geom. Topol. 14 (2010) 2151 | DOI

[17] P Lambrechts, I Volić, Formality of the little N–disks operad, to appear in Mem. Amer. Math. Soc.

[18] J L Loday, B Vallette, Algebraic operads, 346, Springer (2012) | DOI

[19] T Pirashvili, Dold–Kan type theorem for Γ–groups, Math. Ann. 318 (2000) 277 | DOI

[20] D M Pryor, Topological manifold calculus, PhD thesis, University of Virginia (2012)

[21] K Sakai, Configuration space integrals for embedding spaces and the Haefliger invariant, J. Knot Theory Ramifications 19 (2010) 1597 | DOI

[22] D P Sinha, The topology of spaces of knots: Cosimplicial models, Amer. J. Math. 131 (2009) 945 | DOI

[23] V Turchin, Hodge-type decomposition in the homology of long knots, J. Topol. 3 (2010) 487 | DOI

[24] V Turchin, Context-free manifold calculus and the Fulton–MacPherson operad, Algebr. Geom. Topol. 13 (2013) 1243 | DOI

[25] M S Weiss, Embeddings from the point of view of immersion theory, I, Geom. Topol. 3 (1999) 67 | DOI

[26] M S Weiss, Homology of spaces of smooth embeddings, Q. J. Math. 55 (2004) 499 | DOI