Localization of fibrations with nilpotent fibre

Abstract

The theory of localization for nilpotent spaces, parallel to that for nilpotent groups, has proved to be a powerful tool in algébrale topology. Recently, P. Hilton has developed a lo calization theory for relative nilpotent groups (5) and for nilpotent crossed-modules (6) which represents a first step in the construction of a localization theory for nilpotent fibrations and for fibrations with nilpotent fibre (weak nilpotent fibrations). A fibre map f:E —> B is said to be nilpotent if µ1E acts nilpotently on the homotopy groups of the fibre. Nilpotent fibre maps turn out to be the right relativization of nilpotent spaces, because their Moore-Postnikov Systems admit principal refinements [7] . In [8] we constructed a localization theory for nilpotent fibrations. In this note we consider the more general situation of fibrations with nilpotent fibre and develop a localization theory for such fibrations.