The set of multiplicative predifferentials and the rational cohomology algebra of fibre spaces
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.
scientific article; zbMATH DE number 4211073
scientific article