Contemporary Mathematics Localization and Cellularization of Principal Fibrations W. G. Dwyer and E. D. Farjoun Abstract. We prove that any cellularization of a principal fibration is again a principal fibration, and that any localization of a principal fibration with respect to a suspended map is again a principal fibration. The structure group of the new fibration is not necessarily the cellularization (localization) of the original structure group; however, they share the same cellularization (local- ization). 1. Introduction Let Cell A denote the A-cellular approximation functor associated to a pointed space A, and let L f denote the functor given by localization with respect to a map f between pointed spaces (see [1], [5], [7] or 1.1). Given a principal fibration E → X over a connected space X, we show in this note that the induced maps Cell A E → Cell A X and L Σf E → L Σf X are also equivalent to principal fibrations. The appearance of the suspension in L Σf is essential (§4). Let G be the homotopy fibre of E → X, or in other words the group of the principal fibration. It turns out that the fibre of Cell A E → Cell A X is not in general equivalent to Cell A G (but see 2.2). For a simple example of this, let A = S n+1 , so that Cell A is the n-connected Postnikov cover functor (for n = 1, the universal cover functor). If the map π n+1 E → π n+1 X is not surjective, the homotopy fibre of Cell A E → Cell A X has nontrivial homotopy in dimension n, and so this homotopy fibre is not even A-cellular, much less equivalent to Cell A G. Along the same lines, if f is the map S n →∗, then L Σf is the n’th Postnikov section functor. Again, if π n+1 E → π n+1 X is not surjective the homotopy fibre of L Σf E → L Σf X is not equivalent to L Σf G (but see 3.2). Relationship to previous work. The behavior of fibration sequences un- der localization functors and cellularization functors is considered in [1], [3], and [5]. The general conclusion is that localization and cellularization functors pre- serve neither fibration sequences nor principal fibration sequences, although under 2000 Mathematics Subject Classification. Primary 55P60; Secondary 55R05. The first author was partially supported by National Science Foundation grant DMS-0735448. He is also grateful to the Mathematics Department of the University of Bonn for its hospitality while some of this work was being carried out. c 0000 (copyright holder) 1