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Dec 12, 2017 at 15:28 comment added Alexander Körschgen Thank you for the comprehensive answer. I would like to add that there is also a recent preprint by Rezk which computes $Map(BG,BH)$ for 1-truncated compact Lie groups $H$ (see arxiv.org/abs/1608.02999). However, this does not really help with this particular application because one gets $\pi_0 (Map (BG,H)) = \pi_0 (H) = \ast$ for such $H$.
Dec 12, 2017 at 14:49 vote accept Alexander Körschgen
Dec 11, 2017 at 18:50 history answered Nicholas Kuhn CC BY-SA 3.0