Adding Cell complexes

[JoeF131]

Property Suggestion

Most of the concepts below are from the book Algebraic Topology by Allen Hatcher and as such I will only cite the corresponding pages.

I am slightly surprised that cell complexes weren't added to the data base, eventhough the notion of an Topological $n$-manifold is already in the data base. As such I would like to suggest the following properties.

• CW complex (p.5/520)
• Finite dimensional CW complex (p.5)
• Finite CW complex (p.5)
• Simplicial complex (p.107)
• Finite dimensional simplicial complex (p.107)
• Finite simplicial complex (p.107)

Here I write "(...) complex" instead of "has a (...) structure", the same way that "Topological $n$-manifold" actually means "has the structure of an Topological $n$-manifold.

Theorems

For both types of cell complexes, we have the following theorems:

CW complexes

• CW complex $\Rightarrow$ paracompact (p.460)
• CW complex $\Rightarrow$ $T_4$ (p.522)
• CW complex $\Rightarrow$ locally contractible (p.522)
• CW complex $\Rightarrow$ $k_1$-space (p.523)
• CW complex $\land$ compact $\Rightarrow$ finite CW complex (p.520)
• Finite CW complex $\Rightarrow$ compact (p.520)
• Finite CW complex $\Rightarrow$ Embeddable into Euclidean space (p.527)
• CW complex $\land$ Embeddable into Euclidean space $\Rightarrow$ finite dimensional CW complex
• CW complex $\land$ weakly contractible $\Rightarrow$ contractible

Here locally contractible and weakly contractible are both not yet in the data base, but were suggested in #818.

Simplicial complexes

• Simplicial complex $\Rightarrow$ CW complex
• Finite dimensional simplicial complex $\Rightarrow$ finite dimensional CW complex
• Finite simplicial complex $\Rightarrow$ finite CW complex
• Simplical complex $\land$ finite dimensional CW complex $\Rightarrow$ finite dimensional simplicial complex
• Simplical complex $\land$ finite CW complex $\Rightarrow$ finite simplicial complex
• Discrete $\Rightarrow$ finite simplical complex
• Cardinality < $\mathfrak{c}$ $\land$ simplical complex $\Rightarrow$ discrete

Counterexamples and Topological $n$-manifolds.

Despite the amount of theorem, we already have many counterexamples in the data base for the converse of the theorems with the exception of simplical complex $\Rightarrow$ finite dimensional simplical complex, for which I suggest adding the infinite dimensional sphere $S^\infty$ S^\infty and simplical complex $\Rightarrow$ CW complex. I couldn't find a counterexample in the latter case but I found the following results if we additionally assumed our space to be a topological manifold:

• In Dimension 0-3, every topological manifold is a simplicial complex
• In Dimension 4, the $E_8$ manifold (p.86) is a simply connected, compact 4-manifold with no simplicial structure (Proof p.171). It is not known whether there is a 4-manifold without a CW structure
• In Dimension $\geq 5$, every manifold has a CW structure but not necessary a simplicial structure. One such n-manifold has been constructed here and has been proven to have no simplicial structure here. Unfortunately those last two papers are to complicated for me to grasp and as such I don't know how to translate them into a counterexample for the above statement.

Let me know whether I have made any error in my reasoning and whether those concepts should be added to the $\pi$-Base.

[prabau]

I am slightly surprised that cell complexes weren't added to the data base,

FYI, up until very recently, pi-base was mostly reflecting notions of general topology. That is why there is practically nothing about algebraic topology.

Also, things get added gradually when people volunteer to make additions to the database. There are countless notions that are of interest to topologists, even just general topologists, and that are not yet in the database. But it is not very constructive to say: please put all of Engelking in pi-base, or please put all of Hatcher in pi-base. More helpful would be to take much smaller steps that can be more easily reviewed and incorporated. Many baby steps eventually get to the same place, with steady progress along the way.

[JoeF131]

Alright so it is better, then to suggest small general topological properties and find spaces satisfying certain combinations of these properties.

[GeoffreySangston]

It sounds cool to me, but I'm new here and the other contributors who are really putting in the serious labor of operating this website should weigh in (just saw prabau did). Perhaps just start by working on a pull request to add the property which means the space is homeomorphic to a CW complex?

the same way that "Topological n-manifold" actually means "has the structure of an Topological n-manifold.

Topological n-manifod is actually defined in pi-base and in the topology books I'm aware of as a kind of topological space (one which is second countable + Hausdorff + locally Euclidean). On the flip side, see Has a group topology, which means the space is homeomorphic to the underlying space of a topological group. So what about any of the following as the name / aliases?

• Cellular space
• Homeomorphic to a CW Complex (kind of a type error so not my favorite)
• Admits a Cellulation
• Admits a Cell Structure

See cellulation at Wikipedia. Is this exactly the same concept as a cell structure in The Topology of CW Complexes? I think that Hatcher's definition is a bit different. Are these exactly equivalent? I'm sorry, but I just haven't explicitly worked with CW complexes very much.

Perhaps it's worth adding $\mathbb{R}P^\infty$ or $S^\infty$.

[prabau]

There is nothing wrong in what you suggested, but it's not going to be incorporated just like that. We prefer to have a PR (= pull request) for just one property at a time with some associated basic theorems, and then more PRs for more results or more properties, etc. For example, you suggested to add "simply connected" in #851 and it got implemented after a lot of discussion on the best way to present things. That can take a little bit of time.

Would you be interested in contributing PRs yourself?

[JoeF131]

So it would have been better to suggestion "CW complex" as one property without any theorems (or maybe small ones like CW $\Rightarrow$ $T_4$) and instead give one explicite definition or maybe multiple equivalent ones, if I really wanted to add Whitehead's original and now outdated definition.

I've read through the "Contributing" site of the pi-base wiki and watched the yt video and think that it might be fun if I could contribute myself.

[JoeF131]

@GeoffreySangston I mean both $S^\infty$ or $\mathbb{R}P^\infty$ are examples of contractible spaces that are not Baire, which doesn't currently exists in the data base. If we were to add one of both, then I would suggest adding $S^\infty$ since adding $\mathbb{R}P^\infty$ without the existens of any $\mathbb{R}P^n$ is kind of awkward

[prabau]

General guidelines we usually follow when adding a new property is first do a literature search to see if there are any variations in the definition. There often are, with slightly incompatible meanings. So we have a discussion (can be part of an "issue" here) about what the best choice would be. Then a PR would introduce the definition, with references and if needed, a mention of about conflicting definitions that we are not using. The same PR can also contain a few basic theorems relating the new property to other properties. Merging that to pi-base will already show something interesting. Then often we need to illustrate why the converse of a theorem is not true by exhibiting a counterexample. That can be done in a separate PR, usually. Also further PRs to add more theorems.

Some of these things can actually be combined. But the main thing, especially when you start contributing, is to start small, so we can iron out the procedure for working with git/github, etc.

Do you have experience with git/github?

[GeoffreySangston]

Is S103 a Baire space? (I guess so actually since it's hausdorff and compact. But I just wanted to mention that a lot of things are missing for contractible right now. I still think adding $S^\infty$ could be a good idea. Do others agree?) I want to take some time soon to added 'Contractible' to the products of contractible spaces in pi-base, but haven't managed it yet. Clicking the question mark on the Contractible page reveals a lot of other spaces pi-base doesn't yet know about contractibility for. Since 'contractible' was just added, a lot of easy ones aren't accounted for yet.

[prabau]

@GeoffreySangston I mean both S ∞ or R P ∞ are examples of contractible spaces that are not Baire, which doesn't currently exists in the data base. If we were to add one of both, then I would suggest adding S ∞ since adding R P ∞ without the existens of any R P n is kind of awkward

As an example of "starting small", what's the benefit of adding the infinite dimensional versions of these spaces when we don't even have the simple ones? Instead, one could introduce some of the RP^n spaces first, which would more useful to people learning about algebraic topology. It may not illustrate what you want to show, but the other version can come later.

[GeoffreySangston]

Whoops. Implicitly thought $S^2$ was already there. Got carried away with $\mathbb{R}P^\infty$, as you both pointed out in one way or another. The idea though was to add "Has a CW Complex Structure", you want useful examples which aren't covered by topological manifold already.

How about working on $S^2$ then? I think you mentioned it before.

Edit Nov 8: S138 (Countable bouquet of circles) would be distinguished as having a CW complex structure which is not locally Euclidean.

[JoeF131]

@GeoffreySangston You're right that might be a good start. I looked up the properties that were manually added to $S^1$ to see which were relevant to add for $S^2$ and it turns out that pi-base doesn't even know that $S^1$ is not simply connected. I could tomorrow make a list of properties that $S^1$ should additionally satisfy and make another issue, where I talk about them.

[prabau]

For the circle, it may not be very useful to create another issue for that. One can already see what is missing by going to https://topology.pi-base.org/spaces/S000170 and clicking on the rightmost button ("question mark") next to the search box. The two obvious ones to fill in are Contractible and Simply connected. Someone can just do a PR directly for that.

[StevenClontz]

Just chiming in to echo @prabau: if there's community interest in adding these properties, then they would be reasonable to add, but let's focus on one(ish) at a time.

If adding more algebraic topology also means adding more active contributors, I'm especially cool with it. (-: Actually it would be necessary as I personally (at least) will be slower to review such contributions as they're a little outside my wheelhouse.

@JoeF131 please open a more focused issue to discuss where you'd like to begin working towards these goals, and include See #874 so it links back here. I'll close this issue once that's done. Thank you!

[prabau]

@JoeF131 I took at look at what you wrote at the top and it seems a good roadmap of what can be done. Apart from the most esoteric examples maybe, one could start gradually incorporating small pieces of this into pi-base.

[JoeF131]

@StevenClontz I am fairly new to this community and therefore don't know what exactly the community is interested in. As such I could add another issue or just let this be if you'll say that CW-complexes don't really fit into the concept o pi-base.

In the latter case I would still like to contribute though. There are 54 spaces or which pi-base doesn't know whether they're simply connected and/or contractible so correcting this might be fun as well.

[StevenClontz]

I think I like you getting involved with these existing properties first, and once you're sure you want to stick around as an active contributor and have a feel for things, we can revisit this. In that case, let's keep this issue open for now so the discussion isn't set completely aside.

[prabau]

@JoeF131 What is your level of experience with git and github?

[JoeF131]

@JoeF131 What is your level of experience with git and github?

I've worked with github before but the last time was almost 2 years ago, so it might take some time to get back into it. I mean I could simply try adding not simply connected to $S^1$ for the beginning and see how that goes. As long as I work in my own forked repository, there shouldn't really be something that could break

[StevenClontz]

Furthermore, the policy of pi-Base is that if you ever break anything, it's the fault of me and @jamesdabbs - you shouldn't have the ability to break things (at least irrevocably). So be brave! :-)