Property Suggestion: Locally simply connected

[prabau]

We need one or more locally simply connected properties. It is an important concept that has relations at least with simply connected (P200) and semilocally simply connected (P229).

The difficulty is that there are several slightly different versions in the literature.
See https://mathoverflow.net/questions/487326/definition-of-locally-simply-connected-space.

We'll need to decide which versions to add, and choose good names for each. I think it would be beneficial to add all the versions that are in common use in the literature, with discriminating examples to understand the subtleties.

This issue has already been partly discussed as part of #1634. I'll paste below several of the relevant comments from that other issue, and feel free to copy other comments that you deem important.

[mathmaster13]

(Not to sound like a dunce, but why is it an important concept?)

[prabau]

From @GeoffreySangston in #1634 (comment):

Having a local basis of simply connected neighborhoods doesn't seem to be the definition used in practice. E.g. see Wlod AA's comments in Maurizio Barbato's MO thread linked below (made before P_10 got added in an edit). I'm under the impression that at least some topologists/geometers view Borsuk's LC^1 as the proper definition. Also, the wikipedia page cites Hatcher and Munkres for the local basis definition, but I didn't see locally simply connected defined in either when I ctrl+F'd through the texts / checked the index of Hatcher, but maybe I missed them. I vaguely remember lots of discussion about not introducing artificial definitions to pi-base even if they might satisfy a convention.

https://mathoverflow.net/questions/487326/definition-of-locally-simply-connected-space

• This seems to be a very useful resource for the topic of adding locally simply connected space.
• The comment by Denis T also seems highly informative.

Also see here where both Moishe Kohan and Paul Frost state that Borsuk's definition of locally contractible should be considered standard: https://math.stackexchange.com/a/3926424/444923 . It seems surprising that apparently there are ANR's that fail Definition 1 from this page: $Y$ is locally contractible if each $y_0 \in Y$ has arbitrarily small (open) contractible neigborhoods.

[prabau]

From @GeoffreySangston #1634 (comment):

Munkres mentions it at page 495

Okay I see now. He calls it 'locally simple connectedness': "which would require that within each neighborhood of b there should exist a neighborhood U of b that is itself simply connected"
(Worth remembering now that Munkres imposes the convention that neighborhoods are open.)

It seems also worth mentioning that Munkres seems to give this definition off the cuff. It seems like the correct definition should be from a source which treats it formally, like Borsuk's.

[prabau]

From @GeoffreySangston #1634 (comment):

Am I mistaken in thinking that P_8 from Maurizio Barbato's list below is the same as pi-base's second definition of SLSC P229? He does write "P_7 and P_8 are usually taken as the definition of semilocally simply connected space (this notion seems to be by far the most relevant because of its applications in the theory of covering spaces)" so this is just to confirm with others.

https://mathoverflow.net/questions/487326/definition-of-locally-simply-connected-space

It's good that P_10 does imply P_8, according to this list.

I wonder if anybody thinks any of the equivalences of P_8 with P_5, P_6, or P_7 are worth mentioning on the page for P229?

[felixpernegger]

I don't see why allowing us to make a stronger statement alone justifies a property's addition. If it has merit to be added on its own grounds, then that is another story. SLSC is a condition in a well known theorem, at least.

1. locally simply connected is more well-known than many other properties we have (for example see https://scholar.google.de/scholar?start=0&q=%22locally+simply+connected%22&hl=de&as_sdt=0,5)
2. if we add locally contractible (which appears more well-known), it would be odd to leave this out

[GeoffreySangston]

One more comment I think should be copied over (thank you for copying these in!)

#1634 (comment) Also, what would pi-base contributors think about having two separate properties which have distinct names in the main definition title, but which share names in the 'Also known as' synonyms text? That could partially solve this. We choose a distinct name for the main definition, and reserve "Locally Simply Connected" for the 'Also known as text'. It doesn't solve the problem of deciding which definition to actually add first though.

[GeoffreySangston]

(Not to sound like a dunce, but why is it an important concept?)

In addition to felixpernegger's response, I'd add that you can look at Borsuk's book Theory of Retracts and find theorems which use the assumption of $$LC^1$$ (what he calls locally 1-connected); e.g. see Chapter II, Section 8, Homomorphism of Hurewicz. (Beware that Borsuk may impose additional properties on spaces. I am not sure right now and am out of time to check. Edit: I have now had time to check and Borsuk does impose the Hausdorff condition on spaces, on the first page of chapter 1. Luckily this condition is not relevant to the proof of Theorem 16.2, which says that retractions preserve the property of being locally homotopically trivial over $A$, for a space $A$.)

Borsuk's book has a hierarchy of related properties called $$LC^n$$. I think a central fact about his properties is that they are so-called r-invariants (See 17.5). This means they are preserved by r-maps, which are retracts in the sense of category theory (possess a right inverse), and which formally generalize the usual notion of retraction.

I am not a homotopy theorist or very knowledgeable about this, but basically there seems to be a lot of work already studying and using this $$LC^1$$ property. As I understand it, the theory of ANRs is a well-studied topic, and this is a part of that theory.

[prabau]

#1634 (comment) Also, what would pi-base contributors think about having two separate properties which have distinct names in the main definition title, but which share names in the 'Also known as' synonyms text? That could partially solve this. We choose a distinct name for the main definition, and reserve "Locally Simply Connected" for the 'Also known as text'. It doesn't solve the problem of deciding which definition to actually add first though.

That's a good idea. We have done this before, for example arc connected (P95) vs. injectively path connected (P38), which also has "arc connected" as an alias.

[felixpernegger]

I now looked a bit through what is different people mean by "locally simply connected" and it actually varies (and often is just not defined properly).
Maybe it's a good question to ask if it actually matters so much, what exact definition we use (since even if they are not equivalent, they might be for almost all spaces). I'd personally go with Def. (2) of the mathoverflow post, simply because it's the most consistent and easiest. But of course if Borsuk definition is widely used, thats preferable...
Another accessible post about this is: https://wildtopology.com/2023/10/17/local-n-connectivity-vs-the-lcn-property-part-2/
(using different definition (1))

We could also state "Note that different authors mean slightly different things, see mathoverflow" or just check for which property the most (interesting) theorems holds. Especially theorems of the form SLSC + property => Locally simply conncted and Locally simply connected + property => Simply connected would be useful I think.
We can also all locally simply connected after locally contractible etc. if that would help

[GeoffreySangston]

I looked through many textbooks and didn't have much luck finding the stronger versions of 'locally simply connected'. The only explicit reference I saw to the local base version is page 495 from Munkres: "within each neighborhood of $$b$$ there should exist a neighborhood $$U$$ of $b$ that is itself simply connected.". Based on Munkres's convention of neighborhoods being open, this technically corresponds to P1 from Maurizio's list, but is explicitly stated as P2. Maurizio writes, "Actually, P1 is the definition of locally simply connected space accepted by most of the authors." Maybe this is true in papers but not textbooks? I haven't found evidence for this yet (but I can keep looking).

As for as other in-use definitions, Maurizio also writes the following. I'm not sure if these are in common use in the literature or not.

Some authors define a space to be locally simply connected if it satisfies $P_3$ (see e.g. Knapp, Lie Groups and Beyond) or $P_4$ (see e.g. Chevalley, Theory of Lie Groups I), while $P_{10}$ is part of a hierarchy of definitions of locally n-connected spaces

The fact that P_10 is preserved by retractions (and r-maps, which is the same) seems useful. I don't think P_1 or P_2 are preserved by retractions or even deformation retractions. I can't say for sure if this will actually help pi-base, but it seems plausible that not having this makes it difficult to prove some trait.

[prabau]

Not sure if Felix was suggesting to pick just one reasonable definition and go with that one. I haven't looked at all the possible definitions from the Maurizio post yet. But as a general comment, there is nothing wrong in having multiple slightly incompatible properties if they are used in the literature. That would actually be a good thing, as people can then use pi-base as a resource to understand the subtle differences by means of theorems and example/counterexample spaces.

So as pi-base canonical name, we could have "Borsuk-locally simply connected" and "..." (just making that up for now).

[felixpernegger]

I opened #1655 (going in the same direction) which is hopefully easier definition-wise, in case we get stuck on this.

But yes, any reasonable definition for local simply connected with nice theorem would be good by me.