Fibrations with Hopfian properties

Abstract

This study explores the homotopy-theoretic meeting-point of topics in differential topology, combinatorial group theory and algebraic K-theory. The first two are due to H. Hopf and date from around 1930. The third arose in the author's characterisation of plus-constructive fibrations. Let F ( → ί )E →B be a fibration such that i induces an isomorphism of homology with trivial integer coefficients; what is the effect of i on fundamental groups? In particular, when one passes to hypoabelianisations by factoring out perfect radicals, does i induce an epimorphism? Numerous conditions are determined which force an affirmative answer. On the other hand, negative examples of a non-finitary nature are also provided. This leaves the question open in the finitely generated case, where it forms a homological version of the dual to Hopf's original, famous question in group theory. © 1989 The Weizmann Science Press of Israel.