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Let $\theta:G\rightarrow H$ be a morphism of Lie groups.

Given $G$ we have classifying space $BG$ and given $H$ we have classifying space $BH$. This $\theta:G\rightarrow H$ gives a map $B\theta:BG\rightarrow BH$ (I do not fully know how this is done).

Given $G$ we have collection of principal $G$ bundles, denoted by $\text{Prin}(G)$ and given $H$ we have collection of principal $H$ bundles, denoted by $\text{Prin}(H)$.

There is what is called an associated bundle construction which associates for each principal bundle $P\rightarrow M$, a principal $H$ bundle $(P\times H)/G\rightarrow M$.

This gives a map $\text{Prin}(\theta):\text{Prin}(G)\rightarrow \text{Prin}(H)$.

Let $\pi:P\rightarrow M$ be an element of $\text{Prin}(G)$. This is associated to a map $f:M\rightarrow BG$ where pullback of $EG\rightarrow BG$ along $f:M\rightarrow BG$ is the map $P\rightarrow M$.

We already know the map $B\theta:BG\rightarrow BH$. Consider the composition $B\theta\circ f:M\rightarrow BH$. Pullback the principal $H$ bundle $EH\rightarrow BH$ along $B\theta\circ f:M\rightarrow BH$ to get a principal $H$ bundle $Q\rightarrow M$. I am sure this is the same (isomorphic) to the bundle $(P\times H)/G\rightarrow M$.

So, $B\theta:BG\rightarrow BH$ gives $\text{Prin}(\theta):\text{Prin}(G)\rightarrow \text{Prin}(H)$.

Similarly, $\text{Prin}(\theta):\text{Prin}(G)\rightarrow \text{Prin}(H)$ gives $B\theta:BG\rightarrow BH$.

Question : Historically, which construction came first? Is the map $BG\rightarrow BH$ defined before or the associated bundle construction? Suppose $BG\rightarrow BH$ is known before, Is associated bundle construction $\text{Prin}(\theta):\text{Prin}(G)\rightarrow \text{Prin}(H)$ was motivated in the way I mentioned above?.

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    $\begingroup$ I would assume the associated bundle construction, perhaps dating to Cartan, or Ehresmann. Classifying space constructions in general were done by Milnor and Segal, though special cases were definitely known earlier, namely the general linear groups. $\endgroup$
    – David Roberts ♦
    Commented Feb 19, 2019 at 10:24
  • $\begingroup$ @DavidRoberts Can we then say the motivation for $B\theta:BG\rightarrow BH$ came from associated bundle construction? Can you recall seeing this somewhere? Do you know if some place might have something about this? I can read it... $\endgroup$
    – Praphulla Koushik
    Commented Feb 19, 2019 at 10:32
  • $\begingroup$ I don't know, but maybe $\endgroup$
    – David Roberts ♦
    Commented Feb 19, 2019 at 20:29

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