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The rational homotopy type of configuration spaces of two points. (English) Zbl 1069.55006
The authors study the rational homotopy type of the configuration space of two points in a 2-connected closed manifold $$M$$ (compact, without boundary, piecewise linear), $$F(M,2)$$, and prove that its rational homotopy type is completely determined by the rational homotopy type of $$M$$. They exhibit the description of the rational homotopy type of $$F(M,2)$$ in terms of a CDGA-model in the sense of Sullivan’s theory.
The main results of this article are: “for $$M$$ a connected orientable closed manifold with $$H^{1}(M;\mathbb Q)=H^{2}(M;\mathbb Q)=0$$, a CDGA-model of $$F(M,2)$$ can be explicitly determined out of any CDGA-model of $$M$$” and “for $$M$$ a simply-connected closed manifold such that $$H^{2}(M;\mathbb Q)=0$$, the rational homotopy type of $$F(M,2)$$ depends only on the rational homotopy type of $$M$$”.
In the last section, as an application, the authors develop some examples of manifolds admitting a differential Poincaré duality CDGA-model and they study the formality of configuration spaces. They prove that when $$M$$ is 2-connected and closed then $$F(M,2)$$ is formal iff $$M$$ is formal. More generally for any $$k\geq 1$$ and for $$M$$ a simply connected closed manifold, if $$M$$ is not formal then the same is true for $$F(M,k)$$. They also give an example of a non formal space admitting a Poincaré duality CDGA-model. If $$M$$ is a $$S^{5}$$-bundle over $$S^{3}\times S^{3}$$ with non zero Euler class, then there is a Poincaré duality CDGA-model of $$M$$ and $$F(M,2)$$ is not formal.

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