Maps of  B Z=pZ to  BG



                          William G. Dwyer

                       Clarence W. Wilkerson



    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


               ff : 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' identification [Lannes1]
!
!
!!            HomAalg (H (X);H (BZ=pZ)) ss [BZ=pZ; X]
!
!
for!suitable X:
!
!
!
    The research of both authors was partially supported by the NSF, and that o*
 *f the

second author by sabbatical funds from Wayne State University.





    In spite of its limitation to rank one elementary p-groups,Theorem I is exa*
 *ctly

the input needed by the machinery of [Dwyer-Zabro dsky] in the general p-group

case:


Theorem II: If ss is a finite p-group and G is a connected Lie group, then


                   ff : Homgrp(ss; G)=() ! [B ss;BG]


is a bijection, where  denotes equivalence up to G-conjugation.



    Finally,!in!the last section, counterexamples are presented to ff being an *
 *iso-

morphism!for!general finite source groups :
    !
    !
Theorem!III: If the target group GisSU (2),then ff is not surjective if ss is t*
 *he

symmetric!group!of degree 3, and not injective if ss is the cyclic group of ord*
 *er 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 ss 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 :  Bss  ! BG; there exists a homomorphism

'!:!ss! ! G such that f is homotopic to B':
!!!
Notation!!!!The symbol H (  ) will stand for H (; Z=p); A - algwill 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: