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Formality of the little \(N\)-disks operad. (English) Zbl 1308.55006

Mem. Am. Math. Soc. 1079, v, 116 p. (2014).
The little \(N\)-disks operad originated in topology as a means of encoding loop space structures, but its applications have been extensive. In 1999, Kontsevich gave a sketch of the proof that this operad is formal over the real numbers, or that its operad of singular chains with coefficients in \(\mathbb R\) is equivalent to its homology. This result has had a number of applications, ranging from the Deligne Conjecture to work of the authors and others on the homotopy types of spaces of knots. In this manuscript, the authors give a detailed proof of the formality of the little \(N\)-disks operad. Furthermore, their result is stronger than that of Kontsevich in multiple ways. First, their version of the little \(N\)-disks operad includes an operation of arity 0. Second, their formality result is given in the context of commutative differential graded algebras (CDGAs), rather than chain complexes. Third, they also prove formality relatively, by considering the inclusion of the little \(m\)-disks operad into the little \(N\)-disks operad for sufficiently large \(N\).
Because the authors wish to establish formality within CDGAs, a clarification is needed. Sullivan’s functor \(A_{PL}\) from the category of topological spaces to the category CDGAs gives a functor from the category of topological operads to the category of CDGA cooperads; the authors here use the semi-algebraic version \(\Omega_{PA}\). However, applying this functor to each space in a topological operad does not produce a cooperad of CDGAs, since it is not comonoidal. A modification of the definition of a CDGA model for a topological operad is necessary to have the correct framework for the desired result.
The sketch of the proof is as follows. The authors use the Fulton-MacPherson operad \(C[\bullet]\), which is known to be weakly equivalent, as a topological operad, to the little \(N\)-disks operad. Each component space \(C[n]\) is a compactification of the space of normalized ordered configurations of \(n\) points in \(\mathbb R^N\). An important feature of these spaces is that they are compact semi-algebraic manifolds with boundary, and the canonical projection maps \(C[n+k] \rightarrow C[n]\) are semi-algebraic bundles whose fibers are compact manifolds. The functor \(\Omega_{PA}\) is applied to each space \(C[n]\) to obtain a CDGA.
The heart of the proof, then, is the Kontsevich configuration space integral map \(I : \mathcal D(n) \rightarrow \Omega_{PA}(C[n])\) for each \(n \geq 0\), where each \(\mathcal D(n)\) is a CDGA of admissible diagrams. The construction of such diagrams is inspired by relations on differential forms. As \(n\) varies, these CDGAs assemble to form a cooperad \(\mathcal D\) which is shown to be quasi-isomorphic to the cohomology of the little \(N\)-disks operad. Then the map \(I\) is shown to be a quasi-isomorphism, establishing the formality result.
Much attention is given to the precise definitions of all the operads and cooperads involved, and the properties that are used in the proof. The arguments are well-written, and this paper will be useful for anyone wanting to understand this formality result.

MSC:

55P42 Stable homotopy theory, spectra
18D50 Operads (MSC2010)
55P48 Loop space machines and operads in algebraic topology
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[1] V. I. Arnol\(^{\prime}\)d, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227-231 (Russian).
[2] Greg Arone, Pascal Lambrechts, Victor Turchin, and Ismar Volić, Coformality and rational homotopy groups of spaces of long knots, Math. Res. Lett. 15 (2008), no. 1, 1-14. · Zbl 1148.57033 · doi:10.4310/MRL.2008.v15.n1.a1
[3] Gregory Arone, Pascal Lambrechts, and Ismar Volić, Calculus of functors, operad formality, and rational homology of embedding spaces, Acta Math. 199 (2007), no. 2, 153-198. · Zbl 1154.57026 · doi:10.1007/s11511-007-0019-7
[4] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. · Zbl 0285.55012
[5] J. Bochnak, M. Coste, and M.-F. Roy, Géométrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 12, Springer-Verlag, Berlin, 1987 (French). · Zbl 0633.14016
[6] Raoul Bott and Clifford Taubes, On the self-linking of knots, J. Math. Phys. 35 (1994), no. 10, 5247-5287. Topology and physics. · Zbl 0863.57004 · doi:10.1063/1.530750
[7] A. K. Bousfield and V. K. A. M. Gugenheim, On \({\mathrm PL}\) de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. · Zbl 0338.55008
[8] Alberto S. Cattaneo, Paolo Cotta-Ramusino, and Riccardo Longoni, Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol. 2 (2002), 949-1000 (electronic). · Zbl 1029.57009 · doi:10.2140/agt.2002.2.949
[9] Fred Cohen, Cohomology of braid spaces, Bull. Amer. Math. Soc. 79 (1973), 763-766. · Zbl 0272.55012 · doi:10.1090/S0002-9904-1973-13306-3
[10] Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111-118. · Zbl 0136.44104
[11] Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. · Zbl 0961.55002
[12] William Fulton and Robert MacPherson, A compactification of configuration spaces, Ann. of Math. (2) 139 (1994), no. 1, 183-225. · Zbl 0820.14037 · doi:10.2307/2946631
[13] Giovanni Gaiffi, Models for real subspace arrangements and stratified manifolds, Int. Math. Res. Not. 12 (2003), 627-656. · Zbl 1068.32020 · doi:10.1155/S1073792803209077
[14] E. Getzler and J. D. S Jones. Operads, homotopy algebra, and iterated integrals for double loop spaces. Preprint arXiv:hep-th/9403055v1.
[15] Victor Ginzburg and Mikhail Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203-272. · Zbl 0855.18006 · doi:10.1215/S0012-7094-94-07608-4
[16] Thomas G. Goodwillie and Michael Weiss, Embeddings from the point of view of immersion theory. II, Geom. Topol. 3 (1999), 103-118 (electronic). · Zbl 0927.57028 · doi:10.2140/gt.1999.3.103
[17] F. Guillén Santos, V. Navarro, P. Pascual, and A. Roig, Moduli spaces and formal operads, Duke Math. J. 129 (2005), no. 2, 291-335. · Zbl 1120.14018 · doi:10.1215/S0012-7094-05-12924-6
[18] Robert Hardt, Pascal Lambrechts, Victor Turchin, and Ismar Volić, Real homotopy theory of semi-algebraic sets, Algebr. Geom. Topol. 11 (2011), no. 5, 2477-2545. · Zbl 1254.14066 · doi:10.2140/agt.2011.11.2477
[19] Maxim Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 97-121. · Zbl 0872.57001
[20] Maxim Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), no. 1, 35-72. Moshé Flato (1937-1998). · Zbl 0945.18008 · doi:10.1023/A:1007555725247
[21] Maxim Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157-216. · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf
[22] Maxim Kontsevich and Yan Soibelman, Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., vol. 21, Kluwer Acad. Publ., Dordrecht, 2000, pp. 255-307. · Zbl 0972.18005
[23] Pascal Lambrechts, Victor Turchin, and Ismar Volić, The rational homology of spaces of long knots in codimension \(>2\), Geom. Topol. 14 (2010), no. 4, 2151-2187. · Zbl 1222.57020 · doi:10.2140/gt.2010.14.2151
[24] Joseph Neisendorfer and Timothy Miller, Formal and coformal spaces, Illinois J. Math. 22 (1978), no. 4, 565-580. · Zbl 0396.55011
[25] Paolo Salvatore, Configuration spaces with summable labels, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 375-395. · Zbl 1034.55007
[26] Pavol Ševera and Thomas Willwacher, Equivalence of formalities of the little discs operad, Duke Math. J. 160 (2011), no. 1, 175-206. · Zbl 1241.18008 · doi:10.1215/00127094-1443502
[27] Dev P. Sinha, Manifold-theoretic compactifications of configuration spaces, Selecta Math. (N.S.) 10 (2004), no. 3, 391-428. · Zbl 1061.55013 · doi:10.1007/s00029-004-0381-7
[28] Dev P. Sinha, Operads and knot spaces, J. Amer. Math. Soc. 19 (2006), no. 2, 461-486 (electronic). · Zbl 1112.57004 · doi:10.1090/S0894-0347-05-00510-2
[29] Jim Stasheff, What is \(\dots \) an operad?, Notices Amer. Math. Soc. 51 (2004), no. 6, 630-631. · Zbl 1151.18301
[30] Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269-331 (1978). · Zbl 0374.57002
[31] Dmitry E. Tamarkin, Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1-2, 65-72. · Zbl 1048.18007 · doi:10.1023/B:MATH.0000017651.12703.a1
[32] Michael Weiss, Embeddings from the point of view of immersion theory. I, Geom. Topol. 3 (1999), 67-101 (electronic). · Zbl 0927.57027 · doi:10.2140/gt.1999.3.67
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