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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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