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).


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