## 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.

### MSC:

 55R80 Discriminantal varieties and configuration spaces in algebraic topology 55P62 Rational homotopy theory 18D50 Operads (MSC2010)

### Citations:

Zbl 1063.55015; Zbl 1172.13009; Zbl 1152.55004
Full Text:

### References:

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