MATHEMATICAE

Self homotopy equivalences of classifying spaces of compact connected Lie groups by Stefan Ja ckowsk i (Warszawa), James McC lu r e (West Lafayette, Ind.) and Bob O l i v e r (Paris) Abstract. We describe, for any compact connected Lie group G and any prime p, the monoid of self maps BGp ̂ → BGp ̂ which are rational equivalences. Here, BGp ̂ denotes the p-adic completion of the classifying space of G. Among other things, we show that two such maps are homotopic if and only if they induce the same homomorphism in rational cohomology, if and only if their restrictions to the classifying space of the maximal torus of G are homotopic. In an earlier paper [JMO], we gave a complete description of all homotopy classes of self maps of the classifying space BG, when G is any compact connected simple Lie group. In this paper, we extend those results to the case where G is any compact connected Lie group, but only considering self maps of BG which are rational equivalences. Most of the paper deals with self maps of the p-adic completions BGp̂; and the results are extended to global maps only at the end. The first complete description of [BG,BG] for any nonabelian connected Lie group G was given by Mislin [Ms], for the group G = S3. More recently, in [JMO] (and based on earlier work by Hubbuck [Hu] and Ishiguro [Is]), we extended Mislin’s result to a description of [BG,BG] for an arbitrary compact connected simple Lie group G. The assumption that G be sim-ple was, however, crucial: examples were given in [JMO, §7] to show that a similar, simple description of all self maps is unlikely for arbitrary connected G. When G is simple, any f: BGp ̂ → BGp ̂ is either a Q-equivalence or nullhomotopic. The most natural setting for obtaining similar strong results for semisimple or connected groups seems to be to restrict attention to the Q-equivalences. For example, we will see in Corollary 2.6 below that fo