Sections of maps with fibers homeomorphic to a two-dimensional manifold

Abstract

Consider a Serre fibration p :E→B which has constant (up to a homeomorphism) fibers p −1( b), b∈ B. Shchepin's Conjecture. A Serre fibration with a metric locally arcwise connected base is locally trivial if it has a low-dimensional (of dimension n⩽4) compact manifold as a constant fiber. This paper makes a first step toward proving Shchepin's Conjecture in dimension n=2. We say that a Serre fibration p :E→B admits local sections, if for every point b∈ B there exists a section of p over some neighborhood of b. The main result of this paper is the following Theorem 4.4. Let p :E→B be a Serre fibration of LC 0 -compacta with a constant fiber which is a compact two-dimensional manifold. If B∈ANR, then p admits local sections.