Cohomology of 3-points configuration spaces of complex projective spaces
configurations in complex projective spaces
The authors of this paper undertake a careful analysis of the cohomology algebras of \(F(\mathbb CP^m,3)\) and \(C(\mathbb CP^m,3)\), the spaces of ordered and unordered configuration spaces of three points in the \(m\)-dimensional complex projective space, respectively. Among other things, the authors compute the Poincaré polynomials of \(F(\mathbb CP^m,3)\) and \(C(\mathbb CP^m,3)\), give the presentations of the cohomology of these spaces as graded commutative algebras for each \(m\), and provide the structure of the cohomology algebras in the stable case, i.e.~for \(H^*(\lim_m F(\mathbb CP^m,3))\) and \(H^*(\lim_m C(\mathbb CP^m,3))\). The authors also compute the Betti numbers of the subspaces of configurations of three colinear and three non-colinear points in \(F(\mathbb CP^m,3)\), both when \(m\) is finite and for the stable case.NEWLINENEWLINEThe strategy for computing the Poincaré polynomials is to examine the Križ rational model \(E(X,n)\) for \(F(X,n)\), the space of \(n\) configuration points in a smooth complex projective variety \(X\). This model is a bigraded differential algebra and the authors undertake a detailed study of the differential in this algebra upon breaking it up into three summands according to the linearization of the action of the symmetric group. To get the algebra structure, they study the Serre spectral sequences associated to the projections \(F(\mathbb CP^m,3)\to F(\mathbb CP^m,2)\) and \(F(\mathbb CP^m,3)\to F(\mathbb CP^m,1)\) (this is also used in the analysis of the colinear and non-colinear pieces of \(F(\mathbb CP^m,3)\)). For the stable result, the authors use a nice observation that the inclusion \(F(\mathbb CP^\infty,n)\to (\mathbb CP^\infty)^n\) is a homotopy equivalence.
scientific article; zbMATH DE number 6367099