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In the article The Cohomology of Classifying Spaces of H-Spaces by M. Rothenberg and N. Steenrod (https://projecteuclid.org/euclid.bams/1183527356) it is stated as a theorem that if $G$ is a topological group then any $G$-resolution $X$ is a principal $G$-bundle over $B_{G} = X /G$ with the action of $G$ in $X$ as a principal map.

In the article no proofs are given so I would like to know of any reference that would point me into a direction to prove this statement.

A $G$-resolution is defined as a filtered $G$-space $X = \cup_{i=0}^{\infty} X_{i}$ such that it satisfies:

  1. Acyclicity condition: every level $X_{n}$ is contractible in $X_{n+1}$ to a point $x_{0}$ on the bottom $X_{0}$.

  2. Free condition: for every $n$ there is a closed subspace $D_n$ between the levels $X_{n-1}$ and $X_{n}$ such that the action induces a relative homeomorphism $(D_{n}, X_{n-1}) \times G \rightarrow (X_{n}, X_{n-1})$.

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    $\begingroup$ I can't find a reference (there are some mimeographed notes by Rothenberg and Steenrod of the same name, but they don't seem to be online) so I'll just leave a comment. The definition of $G$-resolution seems to be modeled on Milnor's construction of the classifying space using joins. The proof they had in mind is probably a generalisation of Milnor's proof in "Constructions of Universal Bundles, II". In particular, rather than using 2. to show that the action is free and topologically nice, try using 2. to show that the projection is locally trivial. $\endgroup$
    – Mark Grant
    Commented Feb 23, 2018 at 22:11
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    $\begingroup$ As @MarkGrant pointed out the free condition is indeed the one that guarantees local triviality. The key point is to notice that every orbit of a point in $X_{n} - X_{n-1}$ has a unique representative in $D_{n} - X_{n-1}$. This way the trivializing cover would be the open sets of the form $p(X_{n} - X_{n-1})$. In my opinion the definition of a $G$-resolution becomes clearer when thinking of Milgram's classifying spaces. It also would be nice to know where one could find the mimeographed notes $\endgroup$
    – Jrnm
    Commented Feb 25, 2018 at 20:42

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