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Stub for the Milnor construction of universal principal $G$-bundles and the disambiguation page universal bundle (we should have universal vector and associated fibre bundles, not only principal and higher principal cases; and the discussion that there is no difference in classification of locally trivial bundles in the principal versus fibre bundle case if only the effective actions on typical fibers are considered). Added a redirect James Stasheff at Jim Stasheff required at Milnor construction (before the link James Stasheff did not work though many of his papers written under that version of the name are quoted in $n$Lab!).
In general, both the join notation and the link to the “joins” page within this “Milnor construction” page are mathematically incorrect.
If $X$ and $Y$ are compact Hausdorff spaces, it does indeed follow from the tube lemma that the quotient-topology join $X \ast Y$ equals the coarser Milnor join $X \circ Y$. Milnor carefully defined the latter in his 1956 article, as his application was to construct a model of $E G$ for general topological groups $G$.
However, for example $\mathbb{R} \ast pt$ is strictly finer than $\mathbb{R} \circ pt$. That is, the special case of cones are different when dealing with noncompact spaces!
The quotient-topology join is used amongst algebraic topologists, because it stays in the category of CW complexes. The Milnor join is popular amongst general/geometric topologists, because it stays in the category of metrizable spaces.
My recent answer in MathOverflow may be useful to the author of this “Milnor construction” page. One possible way of fixing this error is to amend the “join” page to include the Milnor variant with his $\circ$ notation, and then to refer specifically to that variant from here. I can be enlisted to help by the authors as needed.
Thanks. I have changed “topological group” to “compact topological group” in the entry.
But it remains a stub and I don’t have the intention to expand on it at the moment.
If you would indeed have the energy to edit the entry and improve on it, that would be most welcome!
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