Spectral sequences in analytic homotopy theory (Q921659)

Sullivan's theory of rational homotopical type consists in the comparison of a manifold \(M^ n\) with a certain free differential algebra \({\mathcal A}_{{\mathbb{Q}}}\) over \({\mathbb{Q}}\), and also the isomorphism class of \({\mathcal A}_{{\mathbb{Q}}}\) is an invariant of the \({\mathbb{Q}}\)-homotopical type of \(M^ n\). But though the algebra \({\mathcal A}_{{\mathbb{Q}}}\) is constructed as an abstract free algebra, its construction process is not effective. \textit{S. P. Novikov} [Dokl. Akad. Nauk SSSR 283, 1088-1091 (1985; Zbl 0603.55010); Usp. Mat. Nauk 39, No.5, 97-106 (1984; Zbl 0619.58002)] examined the problem of analytic homotopy theory: realize \({\mathcal A}_{{\mathbb{Q}}}\) as a \({\mathbb{Q}}\)-subalgebra of \(\Lambda^*(M^ n\times {\mathbb{R}}^{\infty})\); in the same paper an effective construction process (*) is suggested in which possibly, however, obstructions may occur. The author proves the following affirmations: 1. For every simply connected manifold \(M^ n(\hat M)\), representing a homotopy type \((\hat M=M^ n\times {\mathbb{R}}^{\infty})\), there exists a construction process (*) for \({\mathcal A}_{{\mathbb{Q}}}\subset \Lambda^*(M)\), in which obstructions do not occur. 2. The homotopy classes of inclusions \(\phi\): \({\mathcal A}_{{\mathbb{R}}}={\mathcal A}_{{\mathbb{Q}}}\otimes {\mathbb{R}}\to \Lambda^*(M)\) with homological fixed form of the closed generators of \({\mathcal A}_{{\mathbb{R}}}\) form a group G, \(G=\exp {\mathcal G}\), where \({\mathcal G}\) is a nilpotent Lie algebra. 3. Let \(M^ n\) be a formal manifold. Then there exists a one-to-one correspondence between homotopical inclusions of \({\mathcal A}_{{\mathbb{R}}}\) and \({\mathcal A}_{{\mathbb{Q}}}\) in \(\Lambda^*(\hat M)\). 4. One can consruct a spectral sequence \(E_ r^{p,q}\) for evaluating \({\mathcal G}.\) These propositions were announced in a previous paper of the author [Usp. Mat. Nauk 43, No.2, 147-148 (1988; Zbl 0673.55010)].