Characterisation of plus-constructive fibrations
The author gives a characterisation of a plus-constructive fibration \({\mathcal E}:\) \(F\to E\to^{p}B\) by proving the equivalence of (i) \({\mathcal E}\) is plus-contructive, that is \({\mathcal E}\) induces a fibration \(F^+\to E^+\to B^+\), (ii) the maximal perfect subgroup \({\mathcal P}\pi_ 1(B)\) of \(\pi_ 1(B)\) acts on \(F^+\) by maps (freely) homotopic to the identity, and (iii) \(\pi_ 1(p)\) is \(EP^ 2R\), that is \(\pi_ 1(p)\) preserves perfect radicals, and \({\mathcal P}\pi_ 1(E)\) acts trivially on \(\pi_*(F^+,*)\). As corollaries of this theorem there are two special cases which cover situations frequently occurring in applications of the plus-construction: 1) Let \(F^+\) be a nilpotent space; then \({\mathcal E}\) is plus-contructive iff \({\mathcal P}_{\kappa 1}(B)\) acts trivially on \(H_*(F;Z)\) (Corollary 2). 2) Let \(N\hookrightarrow G\twoheadrightarrow^{\Phi}Q\) be a group extension, and \({\mathcal P}N=1\); then the fibration B\(N\to BG\to BQ\) is plus-constructive iff \(\Phi\) is \(EP^ 2R\) and [N,\({\mathcal P}G]=1\) (Corollary 3).
maximal perfect subgroup of the fundamental group
scientific article; zbMATH DE number 3842751
scientific article