pi-base
Universal covering spaces, semi-locally simply connected, etc...
Discussed in https://github.com/orgs/pi-base/discussions/817
Originally posted by GeoffreySangston October 21, 2024 (I apologize if this has already been discussed. I looked around and couldn't seem to see anything about it. Also, I've been using pi-base pretty consistently since college and think it's an incredibly helpful tool, so I just want to say thanks!)
I don't see the following properties. Is there a good reason for omitting these?
- Contractible
- Locally contractible (on the same page as contractible)
- Strongly locally contractible (on the same page as contractible)
- Simply connected
- Semi-locally simply connected
- Existence of a universal cover (Corollary 82.2 of Munkres says this is equivalent to path connected + locally path-connected + semi-locally simply connected)
- Locally simply connected
- Acyclic
(Edit: I think it's safe to say the standard definitions for the following depend on having a basepoint, since they depend on statements about the higher homotopy groups. So maybe that is a justification for not having them.)
And one could have properties of the fundamental group, in analogy to simply connected. For example, Hatcher has the following on page 49, "It is a theorem of [Shelah 1988] that for a path-connected, locally path-connected compact metric space X , π1(X) is either finitely generated or uncountable."
If these were added, pi-base should also have the Hawaiian earring space. (There does appear to be a controversy around this name. Hatcher uses the descriptive name The Shrinking Wedge of Circles.) And perhaps there should be the rose with infinitely many petals, which comes up when discussing the previous space. And I assume there's a whole canon of counterexamples here. I'd like to see spaces like the pseudocircle, but perhaps that requires being able to make a more complex kind of query than pi-base desires to support (e.g., specifying the homotopy groups in some complex way).
I'm experimenting with my own fork. I've added 'contractible' as a property and 'contractible => path connected' as a theorem.
I guess I see an issue from a pragmatic perspective. Pi-base currently has 59 path connected spaces. I'm not aware of any other property in pi-base which is implied by contractible and not already implied by path connected, though maybe the contributors here are. Is it okay upon first pull-request just to add the contractible property to a few of these (with sources), and let the others get added over time?
Yes, it's perfectly fine, and even recommended, to have a smaller PR that adds the definition with some theorems and maybe an example or two to illustrate. And then add more in separate PRs. A PR with a smaller scope makes it easier to review and easier to merge as well.
One general comment (Note: it's just my personal take on this). It seems fine to me to add properties like contractible and others that can be expressed in a rather elementary way. But properties that depend a lot more on higher order homotopy or homology groups, like acyclic, Eilenberg-MacLane space, aspherical, etc. require a lot more algebraic topology background and machinery. Up to now, pi-base has been concentrating mostly on general topology, not algebraic topology. Do we really want to have algebraic topology become part of pi-base? Maybe, maybe not. It's not for me to decide. @StevenClontz should give direction on this. Also, is there already a repository for algebraic topology equivalent to pi-base somewhere on the web?
On principle, I have no problem with pi-Base modeling any property preserved by homeomorphism that's of interest to researchers. But we need community members who can confidently review contributions that rely on machinery from algebraic topology. I don't know if we're there yet.
https://mathbases.org/ is the best place to find other mathematical databases, and https://code4math.org is a place to connect with mathematicians from other disciplines interested in computing-enabled research.
It should be the case that a path-connected LOTS is contractible iff it is separable (or equivalently embeddable in $\mathbb R$). This can be seen from a categorization of the possible such LOTS and that the long ray isn't contractible (see here for example). I would imagine at least the forward implication is generalizable, but I don't immediately see how.
I would like to add simply connected with the following definition, as it seems to fit in with pi-base better by not involving concepts like fundamental groups and homotopy. I also don't refer to S000170 (Circle) or S000176 (Euclidean Plane R^2) since pi-base defines those up to homeomorphism and I don't see how to elegantly write this definition without referring to a specific circle / disk.
$X$ is path connected, and every continuous map from the unit circle to $X$ is the restriction of a continuous map defined on the unit disk.
This is the first definition on Wikipedia and the second definition on ncatlab. It appears essentially in this form on page 51, Chapter 1.8 of Spanier's Algebraic Topology. Actually, Spanier writes
A space $X$ is said to be n-connected for $n \ge 0$ if every continuous map $f : S^k \to X$ for $k \le n$ has continuous extension over $E^{k+1}$. A $1$-connected space is also said to be simply connected.
(I just noticed Spanier's text allows n-connected spaces which are not path-connected, which is another concern worth discussing. Another edit: Nevermind! I simply missed that the k = 0 case of this definition implies path connected. So this is the same as appears on ncatlab.)
I have a question though (which I ask at the bottom of this comment). The more common definition in the textbooks seems to involve the fundamental group (usually near the fact that fundamental groups at different base-points of path-connected spaces are isomorphic). This is done in Munkres's Topology, tom Dieck's Algebraic Topology, Massey's A Basic Course in Algebraic Topology and Rotman's An Introduction to Algebraic Topology. I don't see this defined in Engelking or Bourbaki's general topology books (though I'm not that familiar with these references). Also, Encyclopedia of General Topology has a more general definition; it defines 'simply connected' to mean 'arcwise connected + trivial fundamental groups', and (I think) an arc there is a more general concept than in pi-base's P000038 (Arc connected).
The text from Munkres is:
A space $X$ is said to be simply connected if it is a path-connected space and if $\pi_1(X, x_0)$ is the trivial (one-element) group for some $x_0 \in X$, and hence for every $x_0 \in X$.
In addition to giving the first definition above (possibly with some stylistic revision), is it better to also include the fundamental group definition (from Munkres)? I don't see any results for 'fundamental group' when I use the search feature on the github.dev page.
(Side remark: FYI, the notion of "arc connected" will see some revision in pi-base. See the last comment in the discussion in #552. The current "arc connected" will be renamed "injectively path connected", and the new "arc connected" will mean homeomorphically arc connected. This will be a better match with the literature.)
pi-base did not mention the fundamental group anywhere before because it had nearly nothing involving algebraic topology.
Now if a notion is usually expressed in terms of the fundamental group (or even higher alg topol concepts), it would be fine and even recommended to phrase it in those terms. At the same time, if there is also a way to express it in simpler terms, we can complement it by giving both formulations. This can be backed up with references where the users can dig further as needed.