  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.
