Let $H$ be a finite $p$-group, and let $G$ be a compact connected Lie group. Then it is well-known that $[BH,BG]\cong Rep(H,G)$, where $BH$ and $BG$ are classifying spaces and $Rep(H,G)$ is the set of representations of $H$ in $G$. What happens if we assume that $H$ is any finite group -- do we also have $[BH,BG]\cong Rep(H,G)$?
$\begingroup$
$\endgroup$
1
-
$\begingroup$ If $G$ is finite, this is true by obstruction theory. Otherwise it is false in general, I think $H=A_4$ and $G=SO(3)$ or something like this provides a counter example, but I haven't found a source yet. $\endgroup$– user43326Commented Jul 26, 2014 at 12:50
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
In "Maps from $B\pi$ into $X$" Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 153, 117–127., Wojtkowiak proves that the natural map $Rep(H,G)\rightarrow [BH,BG]$ is not surjective when $H=\Sigma _3$, $G=U(2)$.
-
$\begingroup$ Perfect! Thank you for the reference. $\endgroup$– userCommented Jul 26, 2014 at 14:52