Maps of BZ=pZ to BG

The purpose of this note is to give an elementary proof of a special case of the result of [Adams, Lannes2, Miller-Wilkerson] characterizing homotopy classes of maps from the classifying space of an elementary p-group into the classifying space of a connected Lie group. Our result states Theorem I: If G is a connected compact Lie group, then the natural map : Homgrp(Z=pZ;G)=() −! [BZ=pZ;BG] is a bijection, where is equivalence up to G- conjugation. The previous proofs relied on Quillen’s description of H(BG;Fp) as an in-verse limit of certain algebras indexed over the elementary p-groups of G: The present proof does not use this description, but has in common with the other proofs the use of Lannes ’ identication [Lannes1] HomA−alg(H(X); H(BZ=pZ)) [BZ=pZ;X] for suitable X: The research of both authors was partially supported by the NSF, and that of the second author by sabbatical funds from Wayne State University. 1 In spite of its limitation to rank one elementary p-groups, Theorem I is exactly the input needed by the machinery of [Dwyer-Zabrodsky] in the general p-group case: Theorem II: If is a nite p-group and G is a connected Lie group, then : Homgrp(;G)=() −! [B;BG] is a bijection, where denotes equivalence up to G-conjugation. Finally, in the last section, counterexamples are presented to being an iso-morphism for general nite source groups: Theorem III: If the target group G is SU(2), then is not surjective if is the symmetric group of degree 3, and not injective if is the cyclic group of order 15. That is, the behavior observed for SL(2; 5) by Milnor and Adams is generic for non-p-groups. Although the proof of Theorem I avoids Quillen’s main result, the study of the action of Z=pZ on the flag manifold G=T is very much in the spirit of [Quillen]. 1. Obtaining Homomorphisms from Maps Fix a prime p, let be a cyclic group of order p, and let G be a compact connected Lie group. The purpose of this paragraph is to prove the following proposition: 1.1 Proposition Given a map f: B −! BG; there exists a homomorphism ’ : −! G such that f is homotopic to B’: Notation The symbol H( − ) will stand for H(−; Z=p); A- alg will denote the category of unstable algebras over the mod p Steenrod algebra A; and [U; V] will stand for the set of homotopy classes of unpointed maps from U to V