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Gave proper reference for (Kieboom 1987).
gave a reference to a proof for the claim that Hurewicz cofibrant closed subspace inclusions are equivalently those whose pushout-product with $\{0\} \hookrightarrow [0,1]$ has a retraction
and added (here) the observation that this implies, for all spaces involved being compactly generated, that k-ified products with a space preserves the Hurewicz cofibration property of subspace inclusions.
I have removed the previous text and diagrams in the Definition-section (here), whose organization and typesetting was awkward to the point of being unreadable, and replaced it with the polished typesetting of the same material that I just added also to the entry homotopy extension property.
For what it’s worth, I have added a little TikZ-diagram hosting that retraction (here)
What’s an actual proof that Lie groups are well-pointed?
One could draw a cute diagram here, showing how “the group grows a spike at its neutral element”. For $S^1$ this still has a reasonable rendering. Maybe later.
re 6: Strøm’s first note on cofibrations (1966: 12) has a characterization of closed cofibrations (theorem 2) as the inclusions of those closed subspaces $A \subseteq X$ for which there exists an open $U \subseteq X$ which is deformable rel $A\subseteq U$ to $A$ in $X$ and for which there is a function $\varphi\colon X\to [0,1]$ with $\varphi^{-1}(0)=A$ and $\varphi^{-1}(1)=X\setminus U$. That shows that every locally Euclidean Hausdorff space is well-pointed at all of its points. (For $x\in X$, take a chart $U\subseteq X$ around $x$, and an open $\varepsilon$-ball $V\subseteq U$ around $x$; there is a function $\varphi\colon U\to[0,1]$ with $\varphi^{-1}(0)=x$ and $\varphi^{-1}(1)=U\setminus V$, which can be extended to $X$ by taking it to be $1$ on the complement of the closed (compact!) $\varepsilon$-ball in $U$ around $x$.)
Thanks for the hints!
Have now added the content of Strøm’s Thm. 2 here, drawing some diagrams to make the zoo of conditions a little more transparent (to my mind, at least).
Will next spell out in the entry the conclusion about Euclidean spaces that you indicate. Is there are citable reference that makes this conclusion explicit?
Thanks. Yes, I had already seen it in Bredon’s book and added the pointer. That seems to be a pretty good book, have only opened it tonight for the first time.
It’s getting a bit late for me, maybe I need to call it quits and continue tomorrow. What I am really after is verifying that PU(ℋ) and friends are well-pointed. It should all be exercise-level checking, but it’s also a little annoying…
Okay, I have now added the example of locally Euclidean spaces here.
By the way, we are not required to have $\phi^{-1}(\{1\}) \,=\, X\setminus U$, just $\phi^{-1}(\{1\}) \,\supset\, X\setminus U$, no? On the other hand, using the open ball as the neighbourhood we would have this equality. Either way it works, it seems to me.
Or maybe I am too tired now, will call it a day.
It’s late here too, so as a note to myself I’ll leave Wikipedia on ANRs and Aguilar–Gitler–Prieto (2002: 99)’s theorem 4.2.15 (a closed embedding $A\embedsin X$ with $X$ an ANR is a closed cofibration iff $A$ is an ANR – which, they remark, includes paracompact Hausdorff manifolds modeled on Banach spaces, cf. p.99, just after exercise 4.2.11). BTW, you do need some kind of separation property on your locally Euclidean space to make it well-pointed at every point; for example the line with two origins is not well-pointed at either origin since real-valued functions can’t tell them apart (Hausdorffness or KC (=every compact set is closed) was a quick fix). I’ll sleep on the possibly strict inclusions :)
Thanks for the pointer to AGP!, I’ll check that out now.
Regarding separation. Thanks, I guess I forgot to check that extension of $\phi_{\vert U}$ by the function constant on 1 be continuous. So what do you appeal to for this to be the case? Tietze?
Hausdorffness is enough (maybe even necessary): the closed ball of radius $1$ is compact and hence closed in (Hausdorff) space as a whole, so its complement is open, and a function which is continuous on each set in an open cover is continuous. (I’ll try and edit that in, if the train wifi allows it)
Ah, I see there are typos left, but you are editing now.
Okay, maybe we are editing cross-purpose now. What is $U$ in your latest version, it seems to be undefined now? I thought with taking $U$ to be any open ball a little larger than the unit ball I was implementing your proof idea. In your latest version $U$ seems undefined. It seems to me if we change “$V$” back to “$U$” and declare it to be a little larger than unity, then we have a proof. No?
Oh, sorry, I see now that you use $U$ for the chart. Could we then swap $U \leftrightarrow V$ so that the notation matches that of the Proposition we are appealing to.
Okay, thanks. I have just slightly edited, making the naming of the chart more explicit and taking the domain of $\eta$ to be $U \times [0,1]$. In fact then it seems that we can just do away with $V$ and the proof runs nicely. Hope I am not overlooking something.
(Sorry if I have been making it more complicated than necessary, thanks again for all your input!)
Next, I have now typed out the argument, via a couple of references, that closed Banach submanifold inclusions into paracompact Banach manifolds are h-cofibrations: here.
That’s great, this gives all the well-pointedness I need for dealing with the universal equivariant PU(H)-bundle. :-)
Looks good to me (and you were right about merely needing an inclusion $X\varsetminus U\subseteq \phi^{-1}(1)$ — I didn’t notice that that’s how Bredon has it). Sorry for interfering with your edits earlier! It’s great to have this stuff written up somewhere :)
My pleasure. Thanks again for your help.
Oh no, I see now I was wrong to think that U(H) is Banach in the strong topology. It is so only in the norm topology. Need to work harder to show that in the norm-topology it’s still well-pointed…
Thanks. Of course that’s also the statement of the proposition that follows (here).
Whoops, I missed that (I saw “weak Hausdorff compactly generated spaces” and assumed the worst 😅). Actually May’s concise course’s page 44 doesn’t state what category it’s working in and leaves the closedness as an exercise, so it might be better to just show that a subspace $A \subseteq X$ whose inclusion map is a cofibration is a retract of a closed subspace $F$ of $X$, which implies that $A$ is closed in $X$ in both categories (since $A$ is the equalizer of $\operatorname{id}_X$ and $r \circ i$ where $r$ is a retraction of $i \colon A \hookleftarrow F$). This follows from the NDR-pair-like characterization of (not necessarily closed) cofibrations, which would have to be written up first…
Sounds good! If you have the energy, it would be a great service to the community if you could make a further edit to this extent. Thanks!
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