The rational homotopy type of configuration spaces of two points

A remarkable DGmodule model for configuration spaces

“…In the present paper we describe a CDGA When k D 2, F.A; 2/ is weakly equivalent to the CDGA-model of F.M; 2/ built in [13,Theorem 5.6], and when M is a complex projective variety then F.H .M I Q/I k/ is equivalent to the Fulton-MacPherson-Kriz CDGA-model of F.M; k/ built in [8] and [11]. We are not able to prove in general that for k 3, F.A; k/ is a CDGA-model of F.M; k/ but at least we can prove that it is an equivariant DGmodule model of it.…”

“…By contrast, a general position argument implies that for a 2-connected closed manifold the configuration space of two points depends only on the homotopy type of the manifold. More generally we have proved in [13] that the rational homotopy type of F.M; 2/ depends only on the rational homotopy type of M , under the 2-connectivity hypothesis, and we have build an explicit model (in the sense of Sullivan) of that configuration space out of a model of M .…”

Rational homotopy models for two-point configuration spaces of lens spaces

“…Lambrechts and Stanley [7] resolved this conjecture for the rational homotopy type of a configuration of two points in a closed two-connected manifold. In [8], Longoni and Salvatore used Lens spaces to provide the first counterexample.…”

“…Longoni and Salvatore [8] show that Conf 2 ( L (7,1) ) is homotopy equivalent to ( ∨ 6 S 2 ) × S 3 and hence is rationally formal and has no non-trivial Massey products. Then they proceed to construct representing submanifolds for the cohomology classes of Conf 2 ( L (7,2) ) and use these representatives to show that there is a non-trivial Massey product in the cohomology of Conf 2 ( L (7,2) ) .…”

“…Then they proceed to construct representing submanifolds for the cohomology classes of Conf 2 ( L (7,2) ) and use these representatives to show that there is a non-trivial Massey product in the cohomology of Conf 2 ( L (7,2) ) . At the end of their paper they suggest that it would be interesting to study the homeomorphism type of all Lens spaces through the rational homotopy type of configuration spaces.…”