Fibrations with Hopfian properties (Q908544)

The following are equivalent: (A) If \(N\hookrightarrow^{\phi}G\twoheadrightarrow G/N\) is a group extension with \(H_*(\phi)\) an isomorphism, then \(\phi_{{\mathcal P}}\) is an epimorphism. (B) If \(F\to^{i}E\to B\) is a fibration with \(H_*(i)\) an isomorphism, then \(\pi_ 1(i)_{{\mathcal P}}\) is an epimorphism. Here \(G_{{\mathcal P}}\) is the hypoabelianization functor of G: \(G_{{\mathcal P}}=G/{\mathcal P}G\), where \({\mathcal P}G\) is a maximal perfect subgroup. It turns out that (A) (and hence (B)) is not correct. However to each class of groups for which (A) holds there corresponds a class of spaces for which (B) is also true.