Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorDavidRoberts
• CommentTimeJul 23rd 2019

Gave proper reference for (Kieboom 1987).

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeOct 11th 2020

Fixed typo.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeSep 14th 2021
• (edited Sep 14th 2021)

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.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 15th 2021

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeSep 15th 2021
• (edited Sep 15th 2021)

I have added a pointer (here) to p. 44 in May’s “Concise AlgTop” for the claim that Hurewicz cofibrations in (weakly) Hausdorff spaces are always closed.

(But May doesn’t spell out the argument either…)

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeSep 17th 2021

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.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 18th 2021
• (edited Sep 18th 2021)

added (here) pointer to a couple places where proof is spelled out that relative CW-complexes are h-cofibrations.

• CommentRowNumber8.
• CommentAuthorGuest
• CommentTimeSep 18th 2021

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 $x$.)

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeSep 18th 2021
• (edited Sep 18th 2021)

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?

• CommentRowNumber10.
• CommentAuthorGuest
• CommentTimeSep 18th 2021
I don’t know one off the top of my head (or after rummaging through my pdf folder for a bit); Bredon (1993: 432) also has Strøm’s theorem 2 (as theorem 1.5, and with a proof of continuity this time), but thinks the well-pointedness of pointed manifolds is obvious (bottom of p. 435, below definition 1.8). I guess that’s an all too common opinion…
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeSep 18th 2021

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.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 18th 2021

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…

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeSep 18th 2021
• (edited Sep 18th 2021)

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.

• CommentRowNumber14.
• CommentAuthorGuest
• CommentTimeSep 18th 2021

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 :)

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

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?

• CommentRowNumber16.
• CommentAuthorGuest
• CommentTimeSep 19th 2021

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)

Anonymous

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

Thanks!

Just for my own benefit I have expanded out your argument a tad more (here).

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

Ah, I see there are typos left, but you are editing now.

2. Looks good! Made the domain of the homotopy $\eta$ explicit, and fixed $U$ being used for two different things (you don’ need an extra $\varepsilon \gt 0$, BTW)

Anonymous

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

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?

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

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.

• CommentRowNumber23.
• CommentAuthorGuest
• CommentTimeSep 19th 2021
Good point, I forgot about that. I’ll leave the editing to you, then :)
• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

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. :-)

• CommentRowNumber25.
• CommentAuthorGuest
• CommentTimeSep 19th 2021

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 :)

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

My pleasure. Thanks again for your help.

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

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…

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

added (here) mentioning of the example of the point inclusion into $PU(\mathcal{H})$

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeSep 20th 2021

• composition of h-cofibs is an h-cofib

• inclusion into a normal space is h-cofib iff it is so with respect to any open neighbourhood

3. Added the fact that cofibrations into a Hausdorff space are closed (which was already stated in the theorem cited).

Anonymous

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeMay 6th 2022

Thanks. Of course that’s also the statement of the proposition that follows (here).

• CommentRowNumber32.
• CommentAuthorGuest
• CommentTimeMay 6th 2022

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…

• CommentRowNumber33.
• CommentAuthorUrs
• CommentTimeMay 6th 2022

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!