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Obstructions to the section problem in a fibration with a weak formal base. (English) Zbl 0891.55025

A space \(X\) is weakly formal if the integral cohomology and the integral cochain algebras are weakly homotopy equivalent as differential graded algebras. Suppose that \(\xi\: E \to X\) is a fibration with connected fibre \(F\) for which \(\pi_1(X)\) acts trivially on \(\pi_*(F)\) and \(H_*(F)\). Suppose further that \(X\) is a polyhedron and weakly formal. The goal of this paper is to analyze the homotopy classes of sections to \(\xi\), that is, homotopy classes of maps \(f\: X\to E\) with \(\xi\circ f = \text{id}_X\). To get a handle on this difficult problem, the author puts restrictions on the spaces involved. These conditions concern the Hurewicz homomorphism \(\pi_i(F) \to H_i(F)\). For example, if the Hurewicz homomorphism is split injective, the conditions are met. The author gives conditions A and B that are far more technical, but considerably weaker than split injectivity. Under these conditions an obstruction is constructed using a generalization of a twisting cochain and a section exists if and only if this obstruction is transgressively trivial. Under conditions A and B, the obstruction theory can be filtered into a sequence of obstructions which, at each stage, reduce to solving a system of linear equations with respect to elements in \(H^i(X; \pi_i(F))\) for \(i < n\). The author gives a list of conditions that imply the more technical condition B. Under other stricter conditions these techniques lead to a complete determination of all homotopy classes of sections (see Theorem 6.3).

MSC:

55S40 Sectioning fiber spaces and bundles in algebraic topology
55R05 Fiber spaces in algebraic topology
55S35 Obstruction theory in algebraic topology
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