The rational homotopy type of configuration spaces of two points.
type: Item
instance of: scholarly article
title: The rational homotopy type of configuration spaces of two points.
zbMATH Open document ID: 1069.55006
author: Pascal Lambrechts
author: Donald Stanley
publication date: 2005-04-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116129
review text: 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.
reviewed by: Corina Mohorianu
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55R80
zbMATH DE Number: 2162418
zbMATH Keywords: rational homotopy type
zbMATH Keywords: configuration space
zbMATH Keywords: Sullivan theory
zbMATH Keywords: CDGA-model
zbMATH Keywords: formal
MaRDI profile type: Publication
OpenAlex ID: W1739904199
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DOI: 10.5802/AIF.2042
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2162418