Is there a topological Bogomolov--Miyaoka--Yau inequality?

The aim of this paper is to consider a possible extension of the Bogomolov–Miyaoka–Yau inequality to differentiable orbifolds. The conjectured extension is related to the Montgomery–Yang problem about circle actions on the 5–sphere and also to the H–cobordism of Seifert fibered 3–manifolds. Related conjectures on algebraic surfaces with quotient singularities which have the same rational homology as the projective plane are also considered. Finally we give such examples which are birational to the projective plane yet have ample canonical class.