CW complex initial PR

[felixpernegger]

CW complexes are fundamental to algebraic topology, see #1655. Since we have locally contractible now, this seems to be ready.

The name is from wikipedia https://en.wikipedia.org/wiki/Cellular_space, though there they mean finite CW complexes (which I dont see a good reason for), if needed we can switch the alias to be the main name.

Still missing:

- Cellular => Locally path connected

• Discrete => Cellular
• Refinement of Theorem Suggestion: Embeddable into R + Locally connected => Locally contractible #1731

And finally that various spaces have a CW structure.

[GeoffreySangston]

I think I was under the impression once that cellular space has inequivalent definitions in textbooks. I'll investigate this and come back later or tomorrow.

[felixpernegger]

I think I was under the impression once that cellular space has inequivalent definitions in textbooks. I'll investigate this and come back later or tomorrow.

I looked around a bit, and besides the Wikipedia entry (with its single, rather obscure source) I didnt find usage of the term "cellular space" (at least in the way we want). I suggest to use "Admits a CW structure" instead?

[felixpernegger]

Once we merge this, #1633 would be great to have; since it allows us to have a weak version of whitehead theorem in pibase

[GeoffreySangston]

Dold's book has in section V.1: Cellular Spaces:

A space together with a cellular filtration is called a cellular space.

A filtration of a space $X$ is an increasing sequence of subspaces $X^n \subset X$. And a filtration is cellular if relative singular homology groups satisfy $H_i(X^n, X^{n-1}) = 0$ for $i \neq n$, and every singular simplex of $X$ lies in some $X^n$.

In section V.2: CW-Spaces he has:

A Hausdorff space together with a CW-decomposition is called a CW-space (originally "CW-complex")

A CW-decomposition of a Hausdorff space $X$ in his book is a set of subspaces of $X$ satisfying 5 properties, presumably equivalent to Hatcher's definition (at least for our purposes).

[prabau]

P240: "cellular" is not a good name. Also "has a CW structure" does not quite work. I don't think people ever talk about "CW structure". They would say instead that a space has the structure of a CW complex. I.e., we really need to have the word "complex" in there.

We could call it just "CW complex"? "Has the structure of a CW complex" is a little too long.

The same space $X$ could have multiple structures of a CW complex matching its topology. But $X$ could still be described as a CW complex, i.e., there exists a description of it by cell attachments, etc.

As far as the explanation goes, we need to spell out what it means in detail in the text, and not just refer to wikipedia.

[prabau]

In section V.2: CW-Spaces he has:

A Hausdorff space together with a CW-decomposition is called a CW-space (originally "CW-complex")

A CW-decomposition of a Hausdorff space X in his book is a set of subspaces of X satisfying 5 properties, presumably equivalent to Hatcher's definition (at least for our purposes).

Hausdorff need not be assumed. It should be a consequence of the definition. (but we can mention it after the definition)

[GeoffreySangston]

There's also the promising looking reference The Topology of CW Complexes by Lundell-Weingram. They also assume Hausdorff so maybe we should just use Hatcher as the main reference, but this book might still be useful.

Spanier looks like he does not assume Hausdorff.

Rotman's book assumes Hausdorff.

[felixpernegger]

P240: "cellular" is not a good name. Also "has a CW structure" does not quite work. I don't think people ever talk about "CW structure". They would say instead that a space has the structure of a CW complex. I.e., we really need to have the word "complex" in there.

We could call it just "CW complex"? "Has the structure of a CW complex" is a little too long.

The same space X could have multiple structures of a CW complex matching its topology. But X could still be described as a CW complex, i.e., there exists a description of it by cell attachments, etc.

As far as the explanation goes, we need to spell out what it means in detail in the text, and not just refer to wikipedia.

I was initially against this, becuase I understand a Cw complex to be a space with a composition, but many sources indeed just say i.e. "a CW complex is a space which can be constructed by..." (i.e. https://ncatlab.org/nlab/show/CW+complex).
So why not keep it simple I guess and just call it CW complex. @GeoffreySangston do you agree?

I'll write the definition of a CW complex up

[GeoffreySangston]

At least when I took algebraic topology, a CW Complex was not just a space which could exhibit some decomposition, but a space with some specific decomposition into cells. So I think we wouldn't call it just CW Complex for the same reason we don't use "Topological Group" for 'Has a Group Topology'. Maybe in practice though people don't care about the specific CW structure. I am not sure.

[prabau]

("CW-structure" is used after all.)
Regarding terminology, Hatcher p. 5 has:
"A space X constructed in this way is called a cell complex or CW complex."

Later on, although he never defines the term explicitly, he uses "CW structure" to mean a choice of $n$-skeletons and cell attachments etc.
Example, on p. 10:
"If X and Y are CW complexes, then there is a natural CW structure on X ∗ Y having ..."
or on p. 12:
"Let X be the union of a torus with n meridional disks. To obtain a CW structure on X, choose ..."
etc.

So for Hatcher, a CW complex is a topological space admitting a CW structure.

---

Encyclopedia of General Topology, p. 472, has:
"A CW-complex (or CW-space) is a Hausdorff space X endowed with a CW-structure, i.e., an increasing
sequence of closed subspaces X0 ⊆ X1 ⊆ · · · of X which satisfy the following conditions: (i) ... (ii) ... (iii) ..."
Because of the word "endowed", this could be construed as meaning a space $X$ together with the CW structure.
Or just a space admitting such a structure?

[felixpernegger]

I added a definition now and changed the name to CW complex.

I realised that the concrete subspace filtration is in fact actually important (cellular approximation needs it), so I added a disclaimer.

[prabau]

@felixpernegger The definition you gave is insufficient and not clear. Someone who does not know the notion should be able to understand it by reading the page.
Some missing things: you don't even say that $X$ is supposed to be the union of the $X_n$. What does "attaching a cell" mean? What is $X_{-1}$? Is the empty set a CW complex? Need more steps in the inclusion chain to understand what is going on. Need to mention that the sequence of skeletons can be either finite or infinite. etc etc etc.

This really needs to be carefully written so that a neophyte could unambiguously understand it by careful reading without having to guess anything.

[prabau]

For a possible better definition: see p. 472 of Encyclopedia of General Topology.

[prabau]

You also added "cell complex" and "cellular complex" as aliases. What are some sources that use each of these aliases with the meaning of CW complex, and with other meanings?

[felixpernegger]

For a possible better definition: see p. 472 of Encyclopedia of General Topology.

The definition I gave is the standard (historical) definition, we dont need to change it.

[felixpernegger]

@prabau FYI I just made an edit to the wiki page, adding the precise def for attaching cells

Thanks. Not quite right, but it's an improvement.

I noticed I missed the "^k", but is anything else still incorrect? If yes Id like to fix it :)

[prabau]

What you added to the wiki has several problems. The $e^k$ are open cells, and closed sets are needed. The $\alpha$ are indices in some unnamed index set, and not the index set itself. And why do you deviate from the notation $e^k_\alpha$? etc.

Really, better and more precise would be something like is written in the pi-base page itself. And also add a ref. to page 5 of Hatcher.

[prabau]

And don't say "equivalence class" when "equivalence relation" is meant.

[felixpernegger]

I dont know (and dont want to look it up now) how to properly add refs, I need to deviate from $e_a^k$ since we consider product topology. I applied the rest
Good otherwise?

[prabau]

It's still kind of confusing because it's supposed to be an explanation for the previous paragraphs, but the notation does not match the previous pararagraphs. Also, the notation you chose for a closed cells clashes with the notation in the theorem further below. And it seems to be confusing the closed "cells" (which live in X) with the closed balls that are the domains of the characteristic maps. The "boundaries" of the cells are NOT the same as the sphere boundaries of the closed balls. etc. etc. etc.
(the respective boundaries need not even be homeomorphic)

Sorry to say, but it's still seems hastily written and a little sloppy.
But you can leave it alone for now. Hopefully someone will fix it at some point.

[felixpernegger]

For all intents and purposes its fixed now. I think its very clear anyways from context (and there isnt a reasonable other way for it to be interpreted in my opinion).

Just general (well-meant adivce): In case you have ever read the Hatcher book (which is very good), many definitions arent explicitly given and many proofs are essentially "by picture" or even a bit handwavy. And this is not a bad thing.

Not every notation in some proposition has to be 100% elaborated all the time, oftentimes that actually does more damage than benefit. For example, (especially given the text before), it is obvious what is meant with $\overline{e}^k$ and no reasonable reader would confuse that.

[StevenClontz]

I've dug through this a little bit myself. I dabbled in (abstract simplicial) complexes in grad school but it's not really my cup of tea. I was going to go by my library to check out Hatcher, but hey, it's here: https://pi.math.cornell.edu/~hatcher/AT/ATchapters.html

Looking in as an outsider to algebraic topology, I'm not sure the definition we have here is super clear, but I also appreciate that sometimes these "natural" constructions can suddenly become a pain to "rigorously" define as we do in general topology. If our goal is to help mathematicians look up spaces that have properties that are widely cared about, perhaps we shouldn't stress too much about defining things perfectly upfront. @GeoffreySangston mentions multiple inequivalent definitions -- as long as our theorems work for any of these definitions, we can always dig in later (for example, I think this is how "compactly generated" eventually became the three k_i properties in our database).

Frankly, I'd be perfectly fine if we simply give an intuitive description for the property and then cite page 5 of Hatcher for the mathematician interested in the formal definition. For me, the reason we define things within the pi-base data is for internal consistency; if we someday have "CW1 complex" and "CW2 complex" properties, we can dig into the technicalities. At some point, a lot of people just want to search for https://topology.pi-base.org/spaces?q=CW+complex%2B%7EMetrizable and get answers (btw we don't get an answer with this PR) (edit: but we don't need to do that here, just noting as an obvious counterexample that would be good to add in the near future -- would the sequential fan work?).

[felixpernegger]

I've dug through this a little bit myself. I dabbled in (abstract simplicial) complexes in grad school but it's not really my cup of tea. I was going to go by my library to check out Hatcher, but hey, it's here: https://pi.math.cornell.edu/~hatcher/AT/ATchapters.html

Looking in as an outsider to algebraic topology, I'm not sure the definition we have here is super clear, but I also appreciate that sometimes these "natural" constructions can suddenly become a pain to "rigorously" define as we do in general topology. If our goal is to help mathematicians look up spaces that have properties that are widely cared about, perhaps we shouldn't stress too much about defining things perfectly upfront. @GeoffreySangston mentions multiple inequivalent definitions -- as long as our theorems work for any of these definitions, we can always dig in later (for example, I think this is how "compactly generated" eventually became the three k_i properties in our database).

Frankly, I'd be perfectly fine if we simply give an intuitive description for the property and then cite page 5 of Hatcher for the mathematician interested in the formal definition. For me, the reason we define things within the pi-base data is for internal consistency; if we someday have "CW1 complex" and "CW2 complex" properties, we can dig into the technicalities. At some point, a lot of people just want to search for https://topology.pi-base.org/spaces?q=CW+complex%2B%7EMetrizable and get answers (btw we don't get an answer with this PR) (edit: but we don't need to do that here, just noting as an obvious counterexample that would be good to add in the near future -- would the sequential fan work?).

This was baaically what I was doing at the beginning of the PR. Nevertheless, the definition we have now is good and we dont need to change it. Furthermore there simply do not exist multiple (important) inequivalent versions of CW complexes.
I think it is generally good to define the things we use in pibase (as long as its somewhat reasonable; I wouldnt wanna define singular homology), this makes it much easier to work with. See the current mess with all the game theoretic properties for example.

The sequential fan is not a CW complex. But S139 is a non metrizable (finite dimensional) CW complex. Maybe somewhat surprisingly, if one adds weak assumotions on a CW complex, they are always metrizable, see the recent issue I have opened.

[prabau]

... I'd be perfectly fine if we simply give an intuitive description for the property and then cite ...

@StevenClontz
First, speaking not about this property or algebraic topology in particular, I am not sure I agree with this departure from what we have been striving to do with pi-base.
In my mind, the strength of pi-base is that its properties have a precise definition and overall pi-base is very reliable, i.e., we can trust the claimed results (both theorems and values for traits). On multiple occasions in the past (I think more at the beginning of pi-base, at least when I joined), some of the properties were not very carefully defined, leading to ambiguity and incorrect results. Relying on a fuzzy intuitive description of properties does not cut it for me.

Furthermore, in my understanding, pi-base is a really wonderful resource which can be used to do a lot of exploration of topology and from which one can learn a lot. But it is not meant to explicitly "teach". So replacing a precise definition with some fuzzy handwaving is not the way to go.

Also, another strength of pi-base is that it allows to study variations of concepts by containing clusters of closely related properties. Without precise definitions, this would not be possible, and we would have a single ambiguously defined mishmash.

That said, in some cases, the precise definitions can be very long and may require quite a bit of preliminary concepts to state. Examples: some of the game theoretic properties, or some from algebraic topology. So it may be expedient to summarize the property and defer to some references for the full understanding. But when it's possible, I think it's worth giving some more details. Especially when these details are tricky and without them the meaning may be subject to interpretation (like for CW complex).
(Note: #1643 wants to see more details for game related properties)

[GeoffreySangston]

@felixpernegger

"Sure lets move it, but I think we should be honest to ourselves that this definitely wont make any difference to anyone ever"

???
Previously I wrote slightly impolite comments, but what "negative attitude" did I write here???
I just think it is a big waste of time to be overly perfectionistic (like order of references, even though i did change that...)
So for what reason did you dislike my comment exactly now even though I implemented both changes that were suggested?

You're basically telling someone that their contribution is a waste of time. I can't really fathom how you think that's a positive attitude and don't feel the need to explain it. There's a positive way of conveying what you are trying to do, but this is not it. I.e., one could discuss how to make PRs go smoother, but I suspect that the answer would likely not satisfy you.

I fixed the Wiki article for all intents and purposes. I clearly spent (and thought about) the most time on this PR, so how do I not care if it is good or not??

I'm not sure how that is clear at all. I kind of doubt that is the case. From my perspective it looks like you opened a PR on a topic that's actually fairly subtle and then another person did a lot of work to try to clarify that.

[GeoffreySangston]

@felixpernegger I apologize for the self-righteous way I made my point, and wish I did it more gracefully, but I still believe what I've said.

[prabau]

Thanks everyone for airing things out. Now I think we can move past this.

Is there anything else that needs to be done for this PR? I am satisfied with the definition page, but may need to take a last look at the rest.

[felixpernegger]

@felixpernegger

"Sure lets move it, but I think we should be honest to ourselves that this definitely wont make any difference to anyone ever"

???
Previously I wrote slightly impolite comments, but what "negative attitude" did I write here???
I just think it is a big waste of time to be overly perfectionistic (like order of references, even though i did change that...)
So for what reason did you dislike my comment exactly now even though I implemented both changes that were suggested?

You're basically telling someone that their contribution is a waste of time. I can't really fathom how you think that's a positive attitude and don't feel the need to explain it. There's a positive way of conveying what you are trying to do, but this is not it. I.e., one could discuss how to make PRs go smoother, but I suspect that the answer would likely not satisfy you.

I fixed the Wiki article for all intents and purposes. I clearly spent (and thought about) the most time on this PR, so how do I not care if it is good or not??

I'm not sure how that is clear at all. I kind of doubt that is the case. From my perspective it looks like you opened a PR on a topic that's actually fairly subtle and then another person did a lot of work to try to clarify that.

I disagree with every single point you made. Nevertheless, its probably better to not further discuss this.

[felixpernegger]

Thanks everyone for airing things out. Now I think we can move past this.

Is there anything else that needs to be done for this PR? I am satisfied with the definition page, but may need to take a last look at the rest.

I'm fine with it as is.

[prabau]

Approving. Leaving it to @GeoffreySangston to merge.

[felixpernegger]

FYI with 172 comments this is now by almost 50 comments the most commented PR in this repo ever lol

@felixpernegger @prabau I realized I have one thing to suggest. I'm just going to do the awkward thing and suggest it.

[GeoffreySangston]

@felixpernegger @prabau Sorry I did not intend to commit it directly. We can reverse this.

Basically I just think it's cleaner to do $n$-cell attaching than $(n+1)$-cell attaching, and this is what our sources do.

And I also removed scare quotes around "boundary".

[felixpernegger]

@felixpernegger @prabau Sorry I did not intend to commit it directly. We can reverse this.

Basically I just think it's cleaner to do n -cell attaching than ( n + 1 ) -cell attaching, and this is what our sources do.

And I also removed scare quotes around "boundary".

As I said I personally prefer it the other way, but have no strong feelings, so lets just keep it like this

[GeoffreySangston]

I disagree with every single point you made. Nevertheless, its probably better to not further discuss this.

I'm happy to discuss whenever if you want to hash this out. I think you're great by the way. I just did not like a few minor things. Should have maintained my composure. My email is probably online but maybe there's a private messaging system somewhere.

[GeoffreySangston]

Alright I'll approve and let @prabau do final approval or dismiss again like I did if I messed something up in the interrim.

[prabau]

I just think it's cleaner to do n -cell attaching than ( n + 1 ) -cell attaching, and this is what our sources do.

I agree it looks slightly cleaner to do n-cell attaching, which I had suggested before. But ok for now.

(never mind, I did not realize this had already been done)