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In response to an email sent to Urs by Andrew Marshall, a slight amendment to good open cover was made in the proof that paracompact manifolds admit good open covers.
I invested some energy into the proof of the refined statement at good open cover: that every paracompact manifold has an open cover such that every finite non-empty intersection is diffeomorphic to an $\mathbb{R}^n$.
While working on this, I was in contact with colleagues, some of which previously doubted that there is a proof, others told me I am silly to invest any time into such a classical statement.
As far as I can see, the statement is a typical folk theorem a full proof of which was not really written down.
Notice that the statement appears as theorem 5.1 in Bott, Tu Differential forms in algebraic topology ! But the proof given there is not complete either. I added more comments on that and more pointers to the literature in the References-section at ball.
But I think the proof that the nLab entry now gives is complete. But try to poke holes into it.
at good open cover I have made the concept of “differentiably good open cover” more explicit by giving it its own definition-paragraph and cross-linking a bit with the relevant propositions
Just out of curiosity, how many differentiably good open covers of non-contractible manifolds have people seen ’in the wild’, written down with coordinates? I know of the 1-sphere…
Given that (good) covers are often used for defining Grothendieck topologies, I would like to define two covers ${\mathcal{C}}$ and $\mathcal{D}$ of a space $X$ to be “weakly equivalent” if there exists a set isomorphism ${\mathrm{GluedPlots}}({\mathcal{C}},X) \simeq {\mathrm{GluedPlots}}({\mathcal{D}},X)$. For the strong concept, say that a cover ${\mathcal{C}}$ is “equivalent” ${\mathcal{D}}$ if one is a refinement of another. Probably, one can prove that two covers are weakly equivalent with they have a common refinement. Furthermore, any cover ${\mathcal{C}}$ can be refined to a cover $\overline{{\mathcal{C}}}$ which is closed under intersection, that is, if $U$ and $V$ are plots of $X$ in $\overline{{\mathcal{C}}}$, then so is their intersect $U\cap V$. (This also makes sense in the “”good” setting.)
The application of all this is that, to check if a presheaf is a sheaf, it suffices to check (that, for $c$ an object of the site, the canonical map from $X(c)$ to ${\mathrm{GluedPlots}}({\mathcal{C}},X)$ is iso) only for good covers ${\mathcal{C}}$ which are closed under intersection.
For a long time there was a statement at good open cover that CW complexes have them, and while I have no particular reason to doubt that, the attempt at proof before seemed inadequate, so I’ve made some alterations and pointed to the literature as best I could in lieu of a proof.
I just noticed that the References-section (here) spoke of referencing “proof” without saying proof of what. Checking the two references (from #8 above), I see that what is meant is proof of existence of what the entry calls differentiably good open covers of paracompact smooth manifolds.
I have now made that explicit, and added pointer also to Thm. 5.1 in Bott & Tu (which glosses over details) and to Prop. A.1 in our arXiv:1011.4735.
Also I see now that rev 38 had expanded the statement of Prop. 2.1 to a duplicate of the earlier material that is now Prop. 2.3.
It’s good to give alternative proofs, but confusing to state the same proposition twice, for then the reader is left wondering if they are missing any implicit fine-print.
But I leave it as is for the time being…
I have now tried to deal with the duplication introduced in rev 38 by re-working the section (“Existence on paracompact smooth manifolds” here) a little more.
Now it
starts out with a remark (here) on the previous folklore and how it got eventually filled with proofs,
then states the Proposition, once,
and then gives the two proofs. (Not sure if these deserve to be two distinct proofs, as they are closely parallel. Of course it doesn’t hurt to say it twice, but if the idea is that the later proof fixes some gap left by the first, then let’s make that explicit.)
I have also uploaded the pdf copy of that translation of the proof of Gonnord & Tosel and merged it together with the link to the MO discussion where it had emerged into a single reference item Gonnord & Tosel 1998.
I expect that the argument of existence of differentiably good open covers readily generalizes to equivariant open covers of smooth $G$-manfiolds, constructed via the equivariant triangulation theorem as in the proof of Thm. 2.11 in Yang 14 (there might be more canonical references that discuss this):
These would be open covers invariant (as sets of subsets) under the $G$-action, with the property that for all suitable subgroups $H \subset G$ their restriction to $H$ fixed loci is a differentiably good open cover.
This seems to be clear, as one just needs to recognize that the construction as in Yang 14, Thm. 2.11 produces on all fixed loci covers by star-shaped domains. But I haven’t gone through it with a fine comb yet. Maybe somebody knows a reference?
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