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some basics at spherical fibration
added pointer to
In more modern language and more generally, $F$-fiber infinity-bundles are classified by the delooping $B Aut(F)$ of the automorphism infinity-group of F. For $F =S^n$ this gives the classifying space for $n$-spherical fibrations.
The canonical inclusion $O(n+1) \hookrightarrow Aut(S^n)$ picks the subspace of twists coming from the J-,homomorphism.
Ok, we could update like that, but why were they using the H-spaces, $G_n$, thatâ€™s I guess what you call $Map(S^{n-1}, S^{n-1})$, rather than $Aut(S^n)$, the monoid rather than the group?
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