Rational category of the space of sections of a nilpotent bundle
type: Item
instance of: scholarly article
title: Rational category of the space of sections of a nilpotent bundle
zbMATH Open document ID: 0715.55009
DOI: 10.1007/BF02566629
author: Yves Félix
published in: Commentarii Mathematici Helvetici
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140216
review text: Let X be a finite nilpotent CW-complex and Y be a nilpotent space; denote by \(Y^ X\) the space of continuous maps of X into Y. The author proves that the Lusternik-Schnirelmann category (LS-category) of \(Y^ X\) is often infinite. Conditions for finiteness deal only with rational invariants and are therefore computable. The main theorem contains as corollary a previous result of \textit{E. Fadell} and \textit{S. Husseini} [Proc. Am. Math. Soc. 107, No.2, 527-536 (1989; Zbl 0676.58019)]: If Y is a 1-connected space of finite LS-category, with \(\tilde H(Y;{\mathbb{Q}})\neq 0\), then the free loop space of Y has finite LS-category. The proof uses the Haefliger's model for a space of sections [\textit{A. Haefliger}, Trans. Am. Math. Soc. 273, 609-620 (1982; Zbl 0508.55019)] and a determination of Gottlieb's elements whose number is less or equal to the LS-category [\textit{Y. Félix} and \textit{S. Halperin}, Trans. Am. Math. Soc. 273, 1-37 (1982; Zbl 0508.55004)].
reviewed by: Daniel Tanré
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55M30
zbMATH DE Number: 4178122
zbMATH Keywords: nilpotent CW-complex
zbMATH Keywords: Lusternik-Schnirelmann category
zbMATH Keywords: free loop space
zbMATH Keywords: space of sections
zbMATH Keywords: Gottlieb's elements
MaRDI profile type: Publication
OpenAlex ID: W1966332754
description: scientific article; zbMATH DE number 4178122