Poincaré duality and Serre fibrations
Recall that a continuous map is a Serre fibration provided it has the covering homotopy property for maps of polyhedra and it is a Serre \(k\)-fibration if it has the property for maps of polyhedra of dimension at most \(k\). The paper presents an analogue in the setting of maps between manifolds of the detection using Poincaré duality of spheres as those manifolds that are homotopically connected through the middle dimension. The main result is: if \(f: M^m\to N^n\) is a proper Serre \(k\)-fibration of manifolds without boundary such that \(m-n\leq k\), then \(f\) is a Serre fibration if \(m= 2k+1\) or \(m= 2k+2\) and \(f\) has no singular points (i.e., no point-inverse \(f^{-1}(y)\) contains an isolated point). A version for manifolds-with-boundary is also included.
scientific article; zbMATH DE number 1083968