The rational homology of function spaces
type: Item
instance of: scholarly article
title: The rational homology of function spaces
zbMATH Open document ID: 0674.55008
DOI: 10.1007/BF01194391
author: S. H. Smith
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: The category DGCA of differential \({\mathbb{Z}}\)-graded commutative algebras over \({\mathbb{Q}}\) is a closed model category. For any object B of finite type and bounded below, the functor \(-{\hat \otimes}B\) has a left adjoint \((-:B).\) Using this functor, the authors prove: Theorem. Let S denote the category of simplicial sets. Let \(X,Y\in obj(S)\) be connected, with X finite and Y fibrant and nilpotent of finite type. Then for any map \(\psi\) : \(X\to Y\) and any weak equivalence \(\beta\) : \(M^*(X)\to B^*(X)\) with \(B^*(X)\) non-negative of finite type, there is an equivalence in ho-DGCA \((M^*(Y):B^*(X))_{\beta \psi^*}\simeq M^*(Map(X,Y)_{\psi})\). This sheds new and interesting light on Haefliger's construction.
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 55Q05
zbMATH DE Number: 4103975
zbMATH Keywords: models for function spaces
zbMATH Keywords: fibrant object
zbMATH Keywords: differential \({\mathbb{Z}}\)-graded commutative algebras over \({\mathbb{Q}}\)
zbMATH Keywords: closed model category
reviewed by: Yves Félix
MaRDI profile type: Publication
cites work: On the Homology Spectral Sequence of a Cosimplicial Space
cites work: On 𝑃𝐿 de Rham theory and rational homotopy type
cites work: Rational Homotopy of the Space of Sections of a Nilpotent Bundle
cites work: Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces
cites work: Rational homotopy theory
description: scientific article; zbMATH DE number 4103975