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Let $G$ be a topological group. The geometric bar construction $BG = B_{\bullet}(pt, G, pt)$ together with $EG = B_{\bullet}(pt,G,G)$ and the map $EG \to BG$ yields the universal principal $G$-bundle at least, when the identity $e \in G$ is a closed cofibration (a condition that is often called well-pointed).

It is claimed in the book by Rudyak "On Thom spectra, Orientability and Cobordism" in theorem 1.65 (iii) that this still holds true, if $G$ is not well-pointed. Is this an error in the book or did I miss something?

I only had the chance to look up one of the references that Rudyak gives. In "Classifying spaces and fibrations" by May, it is stated in theorem 8.2 that $EG \to BG$ is a principal $G$-bundle in case $e \in G$ is non-degenerate basepoint.

I guess my question is:

How bad is the fibration $EG \to BG$ in the case that $G$ is not well-pointed. Does $BG$ still have the right homotopy type / weak homotopy type?

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  • $\begingroup$ I guess you mean theorem 1.65 in chapter IV? $\endgroup$
    – Mark Grant
    Commented Aug 10, 2012 at 12:44
  • $\begingroup$ Not that I know how to answer in that case, but just to be clear, by "topological group", does it mean strict group? $\endgroup$
    – some guy on the street
    Commented Aug 10, 2012 at 13:09
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    $\begingroup$ Looking at Segal's "Classifying spaces and spectral sequences" (section 3) it seems that the bundle map you describe is not locally trivial in general. But perhaps it is when $G$ is compactly-generated weak Hausdorff (a standing assumption in Rudyak's book)? $\endgroup$
    – Mark Grant
    Commented Aug 10, 2012 at 13:15
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    $\begingroup$ The example I would try to work out in detail is the compact group given by an infinite product of copies of $\mathbb Z/2$. You should also keep in mind that there are two possible ways of interpreting $B_\bullet(pt,G,pt)$: one using the usual geometric realization, and one using the fat geometric realization. $\endgroup$
    – André Henriques
    Commented Aug 10, 2012 at 15:09
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    $\begingroup$ @Dan: The infinite join of copies of $G$, $\mathcal{B} G$, has a map down to the infinite join of copies of $*$, which is $\Delta^\infty$. It is different from $BG$, with either the fat or thin realisation (this is all in Segal's paper). $\endgroup$
    – Oscar Randal-Williams
    Commented Aug 10, 2012 at 22:15

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I don't have a real answer but possibly relevant observations and speculations. I imagine $BG$ can be pretty horrible if $G$ is not well-pointed, unless of course one uses the fat realization, when I imagine the fat construction of $EG$ does not give a bundle. One remark is that $G$ is homotopy equivalent to a well-pointed monoid (not group) $G'$, as noted in Remarks 9.3 of Classifying spaces and fibrations. One might try to compare the categories of principal $G$-fibrations and principal $G'$-fibrations. In another direction, for a space to be well-pointed means that it is cofibrant in the Hurewicz (or Strom) model structure on based topological spaces. As will probably be apparent, I haven't really thought about this, but one might ask for a Hurewicz model structure on topological groups for which cofibrant approximation gives a homotopy equivalent well-pointed group.

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