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The classifying space BG of a group G classifies principal G-bundles, in that homotopy classes of maps [X, BG] are naturally identified with isomorphism classes of principal G-bundles P ⭢ X.
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‘Naturally occurring’ $K(\pi, n)$ spaces, for $n \geq 2$
Following up on Dai's answer, one can go a step further since $P U(H)$ is obviously a group. So if we can find a contractible space on which it acts freely, the quotient will be the next level up (na …