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 f, 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). © 1990 American Mathematical Society.