A counterexample to the Lemaire-Sigrist conjecture
type: Item
instance of: scholarly article
title: A counterexample to the Lemaire-Sigrist conjecture
zbMATH Open document ID: 0927.55006
DOI: 10.1016/S0040-9383(98)00010-X
author: Nicolas Dupont
published in: Topology
publication date: 1999-12-08T00:00:00Z
review text: Recall that the Lyusternik-Shnirel'man category of a space \(X\), \(\text{cat} X\), is the least integer \(n\) such that \(X\) can be covered by \(n+1\) open sets each of them contractible in \(X\). The integer \(\text{cat} X\) is clearly less than or equal to the cone length of \(X\), \(\text{cl}(X)\), that is the least integer \(n\) for which there are cofibration sequences \(\Sigma^i Z_i \to X_i\to X_{i+1}\), \(0\leq i<n\), where \(X_n\) has the same homotopy type as \(X\) and \(X_0\) is contractible. O. Cornea proved that \(\text{cat} X\leq\text{cl}(X) \leq\text{cat} X+1\), while Lemaire and Sigrist conjecture that for a simply connected rational space \(\text{cat} X=\text{cl}(X)\). In this paper the author gives the first example of a rational space with \(\text{cl} (X)=\text{cat} X +1\). The proof is a very nice use of differential graded Lie models for rational space. The category of the space \(X\) is three which is the minimal possible value for such an example, because for spaces \(Y\) with \(\text{cat} Y\leq 2\), we have \(\text{cat} Y=\text{cl}(Y)\).
Mathematics Subject Classification ID: 55M30
Mathematics Subject Classification ID: 55P62
zbMATH DE Number: 1307824
zbMATH Keywords: rational homotopy
zbMATH Keywords: Lyusternik-Shnirel'man category
zbMATH Keywords: cone length
Wikidata QID: Q122975063
reviewed by: Yves Félix
MaRDI profile type: Publication
full work available at URL: s0040-9383(98)00010-x
OpenAlex ID: W2094977506
description: scientific article; zbMATH DE number 1307824