Metaproperty suggestion: Preserved by homotopy equivalences

[felixpernegger]

We now have several properties from Algebraic Topology (and hopefully soon will get more), whuch are generally preserved by homotopy equivalences.
To make this more visible, mentioning this as a meta property where applicable would be good I think.

In general, most properties will likely not be preserved by homotopy equivalences.

[prabau]

Not against it, but why don't we wait until we have an example where this is used in the proof of some trait?

[GeoffreySangston]

I was curious about this so I just went ahead and investigated it.

We definitely get the following as homotopy invariants:

• Connected P36
• Path connected P37
• Simply connected
• Semilocally simply connected P229 (Edit: argued on MSE)
• Has open path components P233. Give $\pi_0(X)$ the quotient topology $X \to \pi_0(X)$. Then P233 is equivalent to $\pi_0(X)$ being a discrete space, and homotopy equivalence $X \to Y$ induces a homeomorphism $\pi_0(X) \to \pi_0(Y)$. (Also see Proposition 5 of Brazas' The topology of path component spaces. He uses a property called semi-locally 0-connected which is equivalent to P233 but missing and not aliased. It might be good to add this because we're thinking about also adding semilocally contractible.)
• Has open connected components P234. One can use the same argument as above but instead apply it to the space of connected components. The main fact is that a homotopy of $X \times [0, 1] \to X$ preserves the connected components of $X$.
• Empty P137 of course.
• $\sigma$-connected P189. Edit: Moved this up here. The argument is pretty easy using the characterization in terms of constant maps to $\omega$ with the cofinite topology. I wrote up a MSE question about this but then figured it out quickly. It's pretty trivial but I think I'll ask/answer. See https://math.stackexchange.com/a/5132784/444923.

The following are all implied by contractible, hence are candidates, but I wasn't sure if they were homotopy invariants or not. These can be found by typing 'contractible' into Dan's 'Select subgraph of properties implied by a chosen property:' tool.

• Weakly locally contractible P224
• Weakly locally simply connected P231

I looked through all the properties in pi-base (by partitioning them into properties satisfied by the singleton and properties not satisfied by the singleton) to see if any of the older ones are homotopy invariants. I think none of them are. Maybe one of Dan's tools automates the following process (also I could have cloned the repo and written a quick python script, and I'd kind of like to have a capability for doing things like that), but basically I opened the singleton in one browser window to its valid properties (using the checkmark) and had another window with the formula filter in the right searching e.g. for contractible + ~P229. I quickly incremented the number from P1 through P234, going through all valid properties for the singleton. This is not a perfect test in principle, but it seemed to work almost perfectly anyway. I think the resulting candidates are likely not homotopy invariants though. They may just be underrepresented in pi-base.

• Cozero complemented P61 (This is probably not a homotopy invariant but I don't know the property well.)
• Cardinality less than every measurable cardinal P164 (A product of a measurable cardinal number of real lines provides a counterexample)
• Density $\leq\mathfrak c$ P209 (This is obviously not a homotopy invariant, but I don't know what a useful counterexample would be.)

(I also did the opposite direction, by looking through the smaller list of properties not held by the singleton. This found that there's no contractible Dowker space known to pi-base. I doubt Dowker is a homotopy invariant, but I again don't know the property well.).

[felixpernegger]

(also I could have cloned the repo and written a quick python script, and I'd kind of like to have a capability for doing things like that

I have been using Claude Code for these sort of things recently, and I can really recommend it for this

[felixpernegger]

• Semilocally simply connected P229

do you have some reference for this? I dont quite see it

[GeoffreySangston]

• Semilocally simply connected P229

do you have some reference for this? I dont quite see it

Sorry about that. An argument is on MSE (works for general spaces). I think these "semilocal" properties should always be homotopy invariants. E.g. Fadell pg. 1 shows wlc (=semilocally contractible) is a homotopy invariant, and "has open path components" (same as "semilocally 0-connected") is a homotopy invariant.

Also, I realized that might want to modify the text of SLSC P229. Many references use the alternative definition that $U$ is an open neighborhood, but these are equivalent. I.e. If $N$ is a neighborhood of $x$ such that $\pi_1(N, x) \to \pi_1(X, x)$ is constant, then there exists an open neighborhood $U$ of $x$ in $N$, and functoriality of $\pi_1$ implies $\pi_1(U, x) \to \pi_1(X, x)$ factors through the trivial map $\pi_1(N, x) \to \pi_1(X, x)$. I think we should edit in "(equivalently, an open neighborhood)" into SLSC, as is done in LC P225; no argument needed there I think.

[felixpernegger]

okk thanks

[GeoffreySangston]

I think a homotopy invariance meta-property is compelling just because pi-base can be used as a general reference, but to make it really compelling for deriving traits pi-base may need spaces homotopy equivalent to other spaces in pi-base and satisfying homotopy invariant properties but not satisfying any of the properties with theorems that imply that homotopy invariant property, and these seem hard to come by (e.g. π-Base, Search for Semilocally simply connected + ~Totally path disconnected + ~locally orderable + ~go-space + ~Weakly locally simply connected + ~$LC^1$); though many SLSC, $LC^1$, and weakly locally simply connected traits haven't been filled in yet since we added these recently, so this test might be misleading. Maybe the same question for simply connected is more promising. E.g. the following search returns $S^2$: π-Base, Search for simply connected + ~contractible + ~indiscrete + ~LOTS, and I could imagine $\mathbb{R}^3 \backslash \{0\}$ making its way in one day.

This ended up being very much tangential, but I thought maybe it's worth sharing still: I sort of misremembered a fact about the pseudocircle S213 and thought deriving its traits might be relevant to this discussion, but actually it's only weakly homotopy equivalent to $S^1$ (maybe should be mentioned in pi-base). Weak homotopy invariance is a stronger meta-property, but many "nice" properties won't be weak homotopy invariants: Part of Hatcher's Proposition 4.13 (CW Approximation) says for every space $X$, there exists a CW complex $Z$ and a weak homotopy equivalence $Z \to X$ (all maps $\pi_n(Z, z_0) \to \pi_n(X, x_0)$ are bijections); this is called a CW approximation of $X$. Path connected is the same as $|\pi_0(X)| = 1$, so path connected is a weak homotopy invariant (so is simply connected), but a connected + ~path connected space has a non-path connected CW approximation, which also must not be connected since connected = path connected for CW complexes. Hence connected is not a weak homotopy invariant. Also SLSC and semi-locally 0 connected are not weak homotopy invariants. Contractible is not a weak homotopy invariant because there exist spaces which are weakly contractible but not contractible, and it follows by Whitehead's theorem that weakly contractible implies contractible for CW complexes.

I'm wondering if it's worth mentioning that certain properties with names that might connote "algebraic topology property" aren't homotopy invariants. E.g. $LC$, $LC^1$, locally contractible, locally simply connected, locally path connected, locally connected (maybe not worth mentioning?); there are contractible spaces not satisfying each of these. I think the weakly locally contractible / weakly locally simply connected properties might turn out this way.

(Edit: Got confused here, and this is wrong, but I'll just leave it to serve as an example of maybe the previous point being useful.) I was thinking about how to make this useful for the traits. I think pi-base would need interesting spaces which aren't contractible but which are homotopy equivalent to other spaces already in pi-base. I looked around for examples. The first pair I found which is missing a trait that could be derived like this is Discrete topology on $\mathbb{R}$ S3 and Sum of continuum-many Right Ray Topology on the Reals S82. I.e., Right Ray Topology on the Reals is contractible, so one can argue that S82 and S3 are homotopy equivalent. Because S82 is currently missing locally contractible but S3 is locally contractible, we could use this metaproperty to help fill in at least one trait.

[felixpernegger]

I think a homotopy invariance meta-property is compelling just because pi-base can be used as a general reference

100% agree

This ended up being very much tangential, but I thought maybe it's worth sharing still: I sort of misremembered a fact about the pseudocircle S213 and thought deriving its traits might be relevant to this discussion, but actually it's only weakly homotopy equivalent to S 1 (maybe should be mentioned in pi-base). Weak homotopy invariance is a stronger meta-property, but many "nice" properties won't be weak homotopy invariants: Part of Hatcher's Proposition 4.13 (CW Approximation) says for every space X , there exists a CW complex Z and a weak homotopy equivalence Z → X (all maps π n ( Z , z 0 ) → π n ( X , x 0 ) are bijections); this is called a CW approximation of X . Path connected is the same as | π 0 ( X ) | = 1 , so path connected is a weak homotopy invariant (so is simply connected), but a connected + ~path connected space has a non-path connected CW approximation, which also must not be connected since connected = path connected for CW complexes. Hence connected is not a weak homotopy invariant. Also SLSC and semi-locally 0 connected are not weak homotopy invariants. Contractible is not a weak homotopy invariant because there exist spaces which are weakly contractible but not contractible, and it follows by Whitehead's theorem that weakly contractible implies contractible for CW complexes.

If we "add" homotopy invariant as a metaproperty, why nt add this as well, sure. Especially once we have CW complexes

I'm wondering if it's worth mentioning that certain properties with names that might connote "algebraic topology property" aren't homotopy invariants. E.g. L C , L C 1 , locally contractible, locally simply connected, locally path connected, locally connected (maybe not worth mentioning?); there are contractible spaces not satisfying each of these. I think the weakly locally contractible / weakly locally simply connected properties might turn out this way.

I also think this may be nice. Personally, I'm quite surprised locally simply connected isnt preserved by homotopy equivalences.