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Toral groups and classifying spaces of $p$--compact groups

Kenshi Ishiguro  
Fukuoka University,  Fukuoka  814-0180, Japan


We show converses to some known results for the classifying spaces 
of $p$--toral groups or $p$--compact toral group.  
Suppose $G$ is a compact Lie group.  The following results are included.

(A)  If there is a positive integer $k$ such that
the $n$--th homotopy groups of $(BG)\p$ are zero for all $n \ge k$,
then $(BG)\p$ is the classifying space of a $p$--compact toral group.
(B)  If the canonical map  $Rep(G, K) >>> [BG, BK]$ is bijective 
for any compact connected Lie group $K$, then $G$ is a $p$--toral group.

We will also discuss the conditions of a compact Lie group that
its loop space of the $p$--completed classifying space be a $p$--compact group. 
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