Cohomology of 3-points configuration spaces of complex projective spaces
type: Item
instance of: scholarly article
title: Cohomology of 3-points configuration spaces of complex projective spaces
zbMATH Open document ID: 1305.55009
author: Samia Ashraf
author: Barbu Berceanu
publication date: 2014-11-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.1291
reviewed by: Ismar Volic
Mathematics Subject Classification ID: 55R80
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 13A50
zbMATH DE Number: 6367099
zbMATH Keywords: configurations in complex projective spaces
zbMATH Keywords: Križ model
zbMATH Keywords: representations of symmetric group
MaRDI profile type: Publication
OpenAlex ID: W2962956585
arXiv ID: 1212.1291
DOI: 10.1515/ADVGEOM-2014-0008
published in: advg
review text: 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.
description: scientific article; zbMATH DE number 6367099