Self-maps of Classifying Spaces of Compact Simple Lie Groups

We describe here the set [BG, BG] of homotopy classes of self-maps of the classifying space BG , for any compact connected simple Lie group G. In particular, we show that two maps ƒ , f : BG —• BG are homotopic if and only if they are homotopic after restricting to the maximal torus of G ; or equivalently if and only if they induce the same homomorphism in rational cohomology. In addition, we identify the homotopy types, up to profinite completion, of the components of the mapping space map(BG, BG). The most central concern of homotopy theory is the classification, up to homotopy, of maps between topological spaces. It has long been suspected that maps between classifying spaces provide a particularly favorable special case of this problem, in which explicit results can be expected. In this paper we announce a complete classification of the self-maps of the classifying space BG, when G is any compact connected simple Lie group. When G and T are arbitrary compact Lie groups, then [BG, BY] will denote the set of unbased homotopy classes of maps from BG to BY. It is natural to ask how closely this set is related to the set Hom(G, Y) of homomorphisms from 6 to T. For any inner automorphism a e Inn(T), Ba is homotopic to the identity on BY. It is thus convenient to write Rep(G, T) = Hom(G, Y)/ Inn(T) ; and ask when the map B: Rev(G,Y)-+[BG,BY]. B(p) = Bp is a bijection. When G and Y are both finite (or even discrete), then B is easily seen to be bijective. A much deeper result, due to Dwyer and Zabrodsky [2], says that B is bijective whenever G is a p-group, (and T is compact Lie). This was extended by Notbohm [8] to the case where G is a p-toral group: i.e., where G has toral identity Received by the editors January 31, 1989 and, in revised form August 27, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 55S37; Secondary 55N91, 55R35. The second author was partly supported by an NSF grant. ©1990 American Mathematical Society 0273-0979/90 $1.00+ $.25 per page 65 66 STEFAN JACKOWSKI, J. E. MCCLURE, AND BOB OLIVER component and n0(G) is a /7-group. These theorems depend on the generalized Sullivan conjecture, proven by Miller, Carlsson, and Lannes using different approaches [16, 17, 18]. Our goal here is to apply the theorems of Dwyer-Zabrodsky and Notbohm to study the sets [BG, BG] when G is a connected, compact, simple Lie group. The first important contribution to this was given by Sullivan [11]. For G = SU(n) and k prime to n\, he showed that the A:th power map on BT (T ÇG the maximal torus) can be extended to an "unstable Adams operation" i// : BSU(n) —• BSU(n). Since the kth power map cannot be extended to a homomorphism, B : Rep(G, G) -> [BG, BG] is far from surjective in these cases—but the sets [BG, BG] still have very simple descriptions. Theorem 1. Let G be a compact connected simple Lie group with maximal torus T and Weyl group W. Then, for any pair of maps ƒ , ƒ' : BG —• BG, the following are equivalent: (1) ƒ and f' are homotopic; (2) foBi~f°Bi\BT-+BG{i:T^G)\ (3) / /*(ƒ; Q) = # * ( ƒ ' ; Q ) . Furthermore, the image of the injective restriction map