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Is there some fix that makes the colimit argument work, even if only in certain special cases? If it is not in fact possible to describe the universal bundle in this way, might it still be possible that $E_\infty^q$ or $\text{colim}\,E_n^c$ is a classifying space (i.e. homotopy equivalent, or weak homotopy equivalent, to $E_\infty^c$)? And if all of these approaches are doomed, why does the colimit explanation proliferate?

Is there some fix that makes the colimit argument work, even if only in certain special cases? If it is not in fact possible to describe the universal bundle in this way, might it still be possible that $E_\infty^q/G$ or $(\text{colim}\,E_n^c)/G$ is a classifying space (i.e. homotopy equivalent, or weak homotopy equivalent, to $E_\infty^c/G$)? And if all of these approaches are doomed, why does the colimit explanation proliferate?

In the second approach, I also have the following worry. I only see the proof of the cited fact as holding in the category of pointed spaces, with basepoint-preserving maps and homotopies (otherwise, the point to which we are contracting seems like it could wander badly). This is an issue for the join, where the null-homotopy of $E_n\subset E_n*G=E_{n+1}$ slurps up $E_n$ to a point in $G$ in a non-basepoint-preserving fashion. Am I missing some argument that will make this still work (e.g. a contraction for basepoints not being preserved, or a null-homotopy of $E_n\subset E_{n+1}$ that preserves basepoints)?

Addendum: In the second approach, I also have the following worry. I only see the proof of the cited fact as holding in the category of pointed spaces, with basepoint-preserving maps and homotopies (otherwise, the point to which we are contracting seems like it could wander badly). This is an issue for the join, where the null-homotopy of $E_n\subset E_n*G=E_{n+1}$ slurps up $E_n$ to a point in $G$ in a non-basepoint-preserving fashion. Am I missing some argument that will make this still work (e.g. a contraction for basepoints not being preserved, or a null-homotopy of $E_n\subset E_{n+1}$ that preserves basepoints)?

I was startled to learn was that I was using the wrong topology on the join. Luckily, I have found some great explanations in §5.7 of Brown's Topology and Groupoids (for the topology on the join) and §14.4 of Dieck's Algebraic Topology (for the contractibility of the infinite join). However, I have been unable to reconcile these explanations with the less formal ones that I have seen/heard, which all centered on the classifying space as a colimit.

What I seek here is an understanding of how (if it all) the colimit explanation can be factored into the general Milnor construction. This requires some explanation of the hand-waving that I've heard, which may not be 100% accurate, so please bear with me.

There seem to be two possible avenues for a fix, but I am not sure that either works out. We could try and restrict to special cases where $E_\infty^c=\text{colim}\,E_n^c$, but Tyrone says that this is not necessarily true, even for $G$ compact Hausdorff. Alternatively, we could hope that $E_\infty^q$ or $\text{colim}\,E_n^c$ is still principal numberable, but I have doubts that this is true.

Is there some fix that makes the colimit argument work, even if only in certain special cases?

I was startled to learn that I was using the wrong topology on the join. Luckily, I have found some great explanations in §5.7 of Brown's Topology and Groupoids (for the topology on the join) and §14.4 of Dieck's Algebraic Topology (for the contractibility of the infinite join). However, I have been unable to reconcile these explanations with the less formal ones that I have seen/heard, which all centered on the classifying space as a colimit.

This whole shift was particularly un-nerving for me because the most commonly described classifying spaces are infinite Grassmannians, which are generally understood in terms of colimit topologies. While this certainly works (since the tautological bundle is numerable and the total space is contractible), the lack of colimits in the general construction makes me feel like I've been thinking about things in the wrong way. As such, what I seek here is an understanding of how (if it all) the colimit explanation can be factored into the general Milnor construction. This requires some explanation of the hand-waving that I've heard, which may not be 100% accurate, so please bear with me.

There seem to be two possible avenues for a fix, but I am not sure that either works out. We could try and restrict to special cases where $E_\infty^c=\text{colim}\,E_n^c$, but Tyrone says that this is not necessarily true, even for $G$ compact Hausdorff. Alternatively, we could hope that $E_\infty^q$ or $\text{colim}\,E_n^c$ (which agree for $G$ compact Hausdorff) is still principal numberable, but I have doubts that this is true.

Is there some fix that makes the colimit argument work, even if only in certain special cases? If it is not in fact possible to describe the universal bundle in this way, might it still be possible that $E_\infty^q$ or $\text{colim}\,E_n^c$ is a classifying space (i.e. homotopy equivalent, or weak homotopy equivalent, to $E_\infty^c$)? And if all of these approaches are doomed, why does the colimit explanation proliferate?