## A remarkable DGmodule model for configuration spaces.(English)Zbl 1152.55004

Given a closed $$n$$-manifold $$M$$, denote by $$F(M,k)=\{x_1,\dots,x_k\in M^k$$, $$x_i\not= x_j$$ if $$i\not=j\}$$ the unordered configuration space of $$k$$ particles in $$M$$. It is known that, if $$M$$ is not necessarily simply connected, the homotopy type of $$F(M,k)$$ does not only depend on the homotopy type of $$M$$. On the other hand, if $$M$$ is $$2$$-connected, the homotopy type (resp. more generally the rational homotopy type) of $$F(M,k)$$ depends only on the homotopy type (resp. the rational homotopy type) of $$M$$. The rational version was indeed proved by the authors in a previous paper. For a simply connected manifold, the problem remains open and this is the assumption in the paper under review.
The authors construct a commutative differential graded algebra over the rationals, $$F(A,k)$$, which models the algebra of PL forms on $$F(M,k)$$, $$A_{\text{PL}}(F(M,K))$$, as differential graded modules over $$A_{\text{PL}}(M^k)$$. Moreover, this model is equivariant with respect to the action of the symmetric group $$\Sigma_k$$ on $$M^k$$. The authors remark that the algebra $$F(A,k)$$ is then a promising candidate to be a model (in the sense of Sullivan) of the configuration space.
This interesting algebra is built out of a model $$A$$ of $$M$$ which is itself a Poincaré duality algebra (this is always possible as was shown by the authors in a previous paper). Through a deep understanding of the algebraic description of the complement of an embedding, the algebra $$F(A,k)$$ is obtained from $$A^{\otimes k}$$ by “adding” a suitable exterior algebra and dividing it by the so called “Arnold” and “symmetry” relations.

### MSC:

 55P62 Rational homotopy theory 55R80 Discriminantal varieties and configuration spaces in algebraic topology
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