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We consider the maps between classifying spaces of the form BK x BL -BG. The main theorem shows that if the restriction map on BL is a weak epimorphism, then the restriction on BK should factor through the classifying spaces of the center of the compact Lie group G. An application implies that BG is an H-space (Hopf space) if and only if G is abelian. For a map f: Y -* Z I the set of the homotopy classes of axes, denoted by f (X, Z), consists of all homotopy classes of maps oa: X -* Z such that there is a map (called a pairing) t: X x Y -* Z with restrictions (axes) [UIx -a and ,ujy -f, [16]. If Z is an H-space, then f'(X,Z) = [X,Z]: for the Hmultiplication m: Z x Z -* Z, a pairing of f and oa is given by the composite map m o (ae x f). It follows that, for example, if GC K, and L are compact Lie groups, then f'(K, G) = [K, G] with f: L -* G. In this paper we will study the classifying space 'version, namely f' (BK, BG) with f: BL -* BG. As group theoretical analog indicates, our results will show that few maps in [BK, BG] belong to f 1 (BK, BG) , in general. Other forms of axial maps (H-pairing) are studied in [6]. The main theorem deals with the case that the map f BL -* BG is a weak epimorphism studied in [8]. (We recall the definition in ?2.) An obvious example of a weak epimorphism is given by a map f = Bp induced by a group epimorphism p. The unstable Adams operations {Ik} are also weak epimorphisms. Theorem 1. Let L and G be connected compact Lie groups and let K be a compact (not necessarily connected) Lie group. Iff : BL -* BG is a weak epimorphism, the following hold: (1) If a G f ' (BK, BG), then the map oa factors through BZ(G) up to homotopy, where Z(G) denotes the center of G. (2) Moreover, we have f L (BK, BG) = Hom(K, Z(G)) Taking L = G and f = id (the identity map), our problem asks about possible BK-actions on BG. In fact, this work was motivated by the following result of G. Dula and D. Gottlieb. Received by the editors April 10, 1995 and, in revised form, June 15, 1995. 1991 Mathematics Subject Classification. Primary 55R35; Secondary 55P15, 55P60.

Classifying spaces and homotopy sets of axes of pairings