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The Lambrechts-Stanley model of configuration spaces. (English) Zbl 1422.55031
Let \(M\) be a closed smooth manifold, and let \[ \mathrm{ Conf}_k(M)=\{ (x_1,\ldots, x_k)\mid x_i\not= x_j \, \forall i\not= j\} \] denote the ordered configuration space of \(k\) points in \(M\). It is natural to ask whether the homotopy type of \(M\) determines the homotopy type of \(\mathrm{ Conf}_k(M)\). An example of R. Longoni and P. Salvatore [Topology 44, No. 2, 375–380 (2005; Zbl 1063.55015)] shows that this is not necessarily the case. However, their example is not simply connected, and the question of the homotopy invariance of \(\mathrm{ Conf}_k(-)\) for simply connected closed manifolds remains open.
A model for a simply connected space \(X\) is a commutative differential graded algebra (CDGA) \(A\) quasi-isomorphic to the CDGA of piecewise polynomial forms \(A^\ast_{\mathrm{ PL}}(X)\). If \(M\) is a smooth manifold, then a real model for \(M\) is a CDGA quasi-isomorphic to the CDGA of de Rham forms \(\Omega^\ast_{\mathrm{dR}}(M)\). In [Ann. Sci Éc. Norm. Supér. 41, No. 4, 497–511 (2008; Zbl 1172.13009)], P. Lambrechts and D. Stanley proved that any simply connected closed manifold \(M\) admits a model \(A\) which satisfies Poincaré duality. In [Algebr. Geom. Topol. 8, No. 2, 1191–1222 (2008; Zbl 1152.55004)] the same authors built, out of such a Poincaré duality model, a CDGA \(\mathrm{ G}_{ A}(k)\) that is quasi-isomorphic to \(A^\ast_{\mathrm{PL}}(\mathrm{ Conf}_k(M))\) as a dg-module.
In this paper, the author considers the real homotopy type of \(\mathrm{ Conf}_k(M)\). He shows that for any Poincaré duality model \(A\) of \(M\) and for all \(k\geq 0\), \(\mathrm{ G}_{ A}(k)\) is a model for the real homotopy type of \(\mathrm{ Conf}_k(M)\). This result then implies the real homotopy invariance of \(\mathrm{ Conf}_k(-)\) for simply connected closed smooth manifolds. The author also shows that if the dimension of the manifold is at least \(4\), then his model is compatible with the action of the Fulton-MacPherson operad, assuming the manifold is framed. The proofs are inspired by Kontsevich’s proof of the formality of the little disks operad.
55R80 Discriminantal varieties and configuration spaces in algebraic topology
55P62 Rational homotopy theory
18D50 Operads (MSC2010)
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