The set of multiplicative predifferentials and the rational cohomology algebra of fibre spaces
type: Item
instance of: scholarly article
title: The set of multiplicative predifferentials and the rational cohomology algebra of fibre spaces
zbMATH Open document ID: 0732.55010
DOI: 10.1016/0022-4049(91)90033-X
author: Samson Saneblidze
published in: Journal of Pure and Applied Algebra
publication date: 1991-01-01T00:00:00Z
review text: In this work, the notion of predifferentials, introduced by \textit{N. A. Berikashvili} [Soobshch. Akad. Nauk Gruz. SSR 51, 9-14 (1968; Zbl 0183.516)], is adapted for two problems in rational homotopy theory: the construction of a new model for a Serre fibration and the description of the set of rational homotopy classes \([X,Y]_{{\mathbb{Q}}}.\) First, let H be a commutative differential graded algebra and let X be a topological space, Saneblidze defines the set of multiplicative predifferentials \(D_ k(X,H)\), where k is a field of characteristic zero. Let \(F\to E\to^{\xi}X\) be a Serre fibration such that \(\pi_ 1(X)\) acts trivially on \(H^*(F;k)\); to \(\xi\) the element \(d(\xi)\in D_ k(X,H)\) is assigned functorially. This d(\(\xi\)) is used for the construction of a model of \(\xi\), a model which gives the filtered model of the fibre in the sense of \textit{S. Halperin} and \textit{J. Stasheff} [Adv. Math. 32, 233-279 (1979; Zbl 0408.55009)]. Next, if X and Y are simply connected CW-complexes of finite type, denote by \(H^*=H^*(\Omega Y;{\mathbb{Q}})\) the Hopf algebra of the cohomology of loop spaces. The author defines the set of Hopf predifferentials \(D_{{\mathbb{Q}}}(X,\tilde RH^*)\) and gets a natural bijection \([X,Y]_{{\mathbb{Q}}}\simeq D_{{\mathbb{Q}}}(X,\tilde RH^*)\). The proofs use techniques of predifferentials and perturbation theory.
reviewed by: Daniel Tanré
Mathematics Subject Classification ID: 55P62
zbMATH DE Number: 4211073
zbMATH Keywords: predifferentials
zbMATH Keywords: Serre fibration
zbMATH Keywords: rational homotopy classes
zbMATH Keywords: filtered model
zbMATH Keywords: cohomology
zbMATH Keywords: loop spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4211073