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

Consider a Serre fibration p :E→B which has constant (up to a homeomorphism) fibers p−1(b), b∈B. inʹs 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. aper makes a first step toward proving Shchepinʹs 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 m 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.