Sur l'homotopie rationnelle des espaces fonctionnels
type: Item
instance of: scholarly article
title: Sur l'homotopie rationnelle des espaces fonctionnels
zbMATH Open document ID: 0597.55008
DOI: 10.1007/BF01172155
author: Micheline Vigué-Poirrier
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155165
review text: The author studies the rational homotopy type of the (pointed) function spaces \((Y,y_ 0)^{(X,x_ 0)}\) and \(Y^ X\) where: X is a nilpotent (pointed) space such that there exists \(k\geq 1\) with \(H^ p(X; {\mathbb{Q}})=0\), \(p>k\) and \(H^ k(X; {\mathbb{Q}})\neq 0\), Y is an (m-1)-connected (pointed) space with \(m\geq k+2\). She shows that the rational homotopy Lie algebra of \((Y,y_ 0)^{(X,x_ 0)}\) is isomorphic (as Lie algebra) to \(H^+(X; {\mathbb{Q}})\otimes (\pi_*(\Omega Y)\otimes {\mathbb{Q}})\). She also finds sufficient conditions for the exponential growth of the sequence of Betti numbers. The context is the theory of minimal model [\textit{D. Sullivan}, Publ. Math., Inst. Hautes Étud. Sci. 47, 269-331 (1977; Zbl 0374.57002)]. More precisely, the main tool is the Haefliger's model for \(Y^ X\) [\textit{A. Haefliger}, Lect. Notes Math. 484, 121-152 (1975; Zbl 0316.57009; Trans. Am. Math. Soc. 273, 609-620 (1982; Zbl 0508.55019)] soon used by \textit{K. Shibata} for the study of Gel'fand-Fuchs cohomology [Jap. J. Math., New Ser. 7, 379-415 (1981; Zbl 0525.57025)].
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55P45
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 3962749
zbMATH Keywords: nilpotent space
zbMATH Keywords: space of sections
zbMATH Keywords: rational homotopy type of
zbMATH Keywords: function spaces
zbMATH Keywords: rational homotopy Lie algebra
zbMATH Keywords: growth of the sequence of Betti numbers
zbMATH Keywords: minimal model
reviewed by: Jean-Claude Thomas
MaRDI profile type: Publication
cites work: Formes différentielles et suites spectrales
cites work: Rational Homotopy of the Space of Sections of a Nilpotent Bundle
cites work: Lectures on minimal models
cites work: The Sullivan conjecture on maps from classifying spaces
cites work: On a theorem of Borsuk
cites work: Rational homotopy theory
cites work: Q3038214
cites work: Infinitesimal computations in topology
cites work: Dans le fibre de l'espace des lacets libres, la fibre n'est pas, en général, totalement non cohomologue à zéro
cites work: Q3719493
OpenAlex ID: W12543639
description: scientific article; zbMATH DE number 3962749