INVERSE LIMITS OF PRINCIPAL FIBRATIONS By JOEL M. COHENf [Received 15 March 1972] Given a family of fibre bundles (i? a ), n ^ a > * ne cartesian product of the bundles, is not a fibre bundle, unless almost all of the original bundles are trivial: the product projection is not locally trivial. For homotopy - theoretic purposes, however, the most important property of a bundle is that it is a fibration; i.e. the projection map satisfies the covering homotopy property (CHP). This property is preserved when taking products. The purpose of this paper is to study inverse limits of fibrations (since 'inverse limit' is a generalization of product) and find conditions under which inverse limits of fibrations are fibrations. We introduce the notion of 'big compact Lie groups' (roughly they are inverse limits of finite-dimensional compact Lie groups), and show that an epimorphism of such groups is a fibration. We use this to calculate the homotopy groups of a compact abelian group G, in terms of its Pontrjagin dual §: TT O {G) = Ext(#,Z), TT^G) = Hom(£,Z), and n^O) = 0 when i > 1. The notation is that of the preceding paper, [4], and that paper contains most of the definitions. The author is grateful to the referee, who did much work on this paper and formulated Proposition 1.2. 1. Principal fibrations Let 0 be a topological group and let X be a free (-r-space. This will mean that 0 acts on the right of X and if, for any x e X and g e G, xg — x then g = e, where e is the identity element of G. X is assumed to be a based space with base-point *. X/G will denote the quotient space of orbits; i.e. elements of X/G are the subspaces xG. *G, the orbit of the base-point, is called the fibre of the action. DEFINITION. A principal fibration is a pair (G,X), where G is a topological group, X is a free G-space and the projection X -> X/G has the CHP. Thus *G -» X -> X/G is a fibration. Any principal fibre bundle over a paracompact HausdorfiF space is a principal fibration (cf. Dold, [5]). For example, (G, X) is a principal t This work was partially supported by the National Science Foundation. Proc. London Math. Soc. (3) 27 (1973) 178-192