Space Suggestion: Connected compact locally 1-Euclidean spaces with cut points

[GeoffreySangston]

Space Suggestion

I was working on adding the circle with two origins and I noticed a space matching the title is missing: π-Base, Search for compact + Locally $n$-Euclidean + has a cut point. There seem to be 4 reasonable "simplest" candidates, and I'm equally interested in 1, 2, and 3. I don't think these can be distinguished in pi-base, but since from one perspective they're equally simple/complex, it's not objectively clear which is worth adding.

1. Let $X_1$ be a closed interval with two endpoints on each side, completed to a compact locally $1$-Euclidean space by attaching loops on each end. This looks like a hand-cuff. I.e., $X_1 := ([0, 1] \times \{1, 2, 3, 4\}) /\sim$, where $\sim$ is the minimum equivalence relation such that

• $\langle x, 1\rangle \sim \langle x, 2 \rangle$ for $x \in (0, 1)$,
• $\langle 0, 1 \rangle \sim \langle 0, 3\rangle$ and $\langle 0, 2 \rangle \sim \langle 1, 3\rangle$,
• $\langle 1, 1 \rangle \sim \langle 0, 4\rangle$ and $\langle 1, 2 \rangle \sim \langle 1, 4\rangle$.

(I.e., closed intervals 1 and 2 form an interval with doubled endpoints, and closed intervals 3 and 4 form the attached loops.)

This one is homeomorphic to the quotient space $S^1 / \sim$ of the unit circle $S^1$ by identifying points $\langle x, y\rangle \sim \langle x, -y\rangle$ exactly when $x \in (-\frac{1}{2}, \frac{1}{2})$. This quotient space structure on $S^1$ seems a bit contrived, yet easy to imagine, but for 2 and 3 the analogous thing seems pretty nice.

2. Let $X_2$ be a closed interval such that the ends loop back on themselves like a butterfly proboscis to form a locally Euclidean space. You can set up an equivalence relation on $[0, 1] \times \{1, 2, 3, 4, 5\}$, but I'll spare you the details here. This one has a neat quotient space structure on $\mathbb{R}$. Let $\sim$ denote the minimum equivalence relation on $\mathbb{R}$ such that $x \sim x + 1$ if $x > 1$ and $x \sim x - 1$ if $x < 1$. So this relation is a subset of the equivalence relation on $\mathbb{R}$ induced by the orbits of the translation action of $\mathbb{Z}$ on $\mathbb{R}$.

Alternatively, form $\mathbb{R} / \sim$ by $x \sim 2 x$ if $|x| > 1$. This is a modification of a Hopf manifold construction (the $1$-dimensional case is called a Hopf circle in Goldman's book).

3. $X_3$. Same as last time, except replace the bar between the circles with a single point. I.e., take two telophase topologies $\{0_1, 0_2\}\ \cup (0, 1]$, and $\{0_1', 0_2'\}\ \cup (0', 1']$, and glue $0_1 \sim 1$, $0_1' \sim 1'$, and $0_2 \sim 0_2'$. Can also be defined similarly to $1$ as $([0, 1] \times \{1, 2, 3, 4\})/\sim$ for some $\sim$. This one has an even neater quotient space structure on $\mathbb{R}$ than 2. Let $\sim$ denote the minimum equivalence relation on $\mathbb{R}$ such that $x \sim x + 1$ if $x > 0$ and $x \sim x - 1$ if $x < 0$. The Hausdorff quotient map has a fiber with 3 points, and the others with $1$, whereas for the other spaces it has two fibers with $2$ points and the others with $1$.

(The multiplicative analog suggested by 2 would seem to be $\mathbb{R} / \sim$, where $x \sim 2 x$ for all $x$, but this is not homeomorphic to $X_3$. Instead, I think it's homeomorphic to $S^1 \sqcup S^1$ extended by a focal point. Funnily enough, I mentioned adding this space at the bottom of my other suggestion https://github.com/pi-base/data/issues/1051, as this is also the leaf space of the Reeb foliation on $S^3$. I asked a professor here if there's any deeper reason to this coincidence and they said they think no and that it's just very common that this focal point extension should appear.)

4. And for this one the idea is to have an interval with a loop on one end, as in 1, and have the other end loop back on itself, as in 2. I don't really expect this one to be added, but the description in terms of $[0, 1] \times \{1, 2, 3, 4, 5\}$ is equally simple/complex as the any of the previous ones, so it seems worth mentioning.

Rationale

Each of these has a cut point, which might be surprising since compact Hausdorff locally Euclidean spaces don't have cut points: π-Base, Search for compact + hausdorff + locally Euclidean + has a cut point

[GeoffreySangston]

There are spaces related to these but slightly more complicated which satisfy π-Base, Search for compact + Locally $n$-Euclidean + Has a cut point + ~arc connected (actually ~locally path injective).

E.g., cap off the branching line https://github.com/pi-base/data/issues/1011#issue-2718994025 with loops (i.e., the construction from 1) and/or "curling back loops" (i.e., the construction from 2). These are not injectively path connected for the same reason as telophase topology S65 or branching line.