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.


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