Rational homotopy models for two-point configuration spaces of Lens spaces
The configuration space of \(k\) points in a manifold \(M\) is the space \[ \text{conf}_k(M) = \{(x_1, \dots, x_k)\in M^k | x_i\neq x_j , \text{for } i\neq j\},. \] A long standing conjecture was that two homotopy equivalent compact manifolds of the same dimension would have homotopy equivalent configuration spaces. In 2005, \textit{R. Longoni} and \textit{P. Salvatore} have proved that this is not the case: Denote by \(L(p,q)\) the usual Lens spaces, then \(L(7,1)\) and \(L(7,2)\) are homotopy equivalent but their configuration spaces with two points are not homotopy equivalent [Topology 44, No.~2, 375--380 (2005; Zbl 1063.55015)]. In this paper using intersection theory, M. Miller makes a deep study of the homology of the universal cover of the spaces \(conf_2(L(p,q))\). The result is a lot of information concerning the space \(\widetilde{conf}_2(L(p,q))\). The text contains the description of its integral cohomology, the list of all its integral non-trivial Massey products, and a rational model.
scientific article; zbMATH DE number 5950680