Sections of maps with fibers homeomorphic to a two-dimensional manifold
The paper provides a partial answer to the question whether a fibration \(p: E\to B\), having constant (up to homeomorphism) fibres \(p^{-1}(b)\), \(b\in B\), is locally trivial. It was conjectured (by Shchepin) that this holds provided \(p\) is a Serre fibration, the base \(B\) is locally path connected and the (typical) fibre is a compact at most 4-dimensional manifold. The main result of the paper states that if \(p:E\to B\) is a Serre fibration of \(LC^0\)-compact metric spaces, the constant fiber is a compact 2-dimensional manifold and \(B\) is an absolute neighborhood retract, then \(p\) admits local sections, i.e. for each \(b\in B\), there is a section of \(p\) over a neighborhood of \(b\). The proofs make use of the quite involved technique of graph approximations to set-valued maps.
scientific article; zbMATH DE number 1760997