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I was playing with some ideas in a vague way and I have encountered this structure that arises from taking the space of angles $\mathbb{S}^1$ and quotienting it by the relation $(x, 360-x)$ (here $360$ refers to degrees and can be $2\pi$ if you want to work in radians).

This is a very similar looking structure to the projective plane $\mathbb{RP}^1$ which arises from $\mathbb{S}^1/(x \rightarrow x + 180) $ however its worth noting that $360-x$ has a fixed point $x=0$ and $x+180$ does NOT have any fixed points so this is definitely a different structure.

My friend and I have been calling this "the arc" but I figured to ask if anyone knows what this object is and what sort of algebraic structures can be naturally associated with it? It's not even clear if it is a group at all but on very small slices it seems to behave like $\mathbb{S}^1$.

JonathanZ
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Sidharth Ghoshal
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    As John Hughes says, this is "the closed unit interval" or "a closed line segment". Some comments - you can picture what's happening as taking a circle and squashing it from two sides to make it flat (gluing the points from one side to the other). For algebraic structure - it cannot be made into a topological group, because its automorphism group doesn't act transitively on it. Addition mod 1 is not an algebraically or topologically very nice operation on the closed interval! – Izaak van Dongen May 09 '24 at 17:30
  • Thanks for commenting! Can this have some kind of type that is not "topological group" but more specific than "arbitrary group on geometric set". I claim that this arc is a "nice" or "natural" structure yet I do not have a name for what the "type" of this group + underlying set should be. – Sidharth Ghoshal May 09 '24 at 17:34
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    What is the nice/natural structure you're talking about? The operation of addition of angles modulo $360$ doesn't descend naturally to this quotient. But even considering addition mod $1$ on $[0, 1]$, there is no identity element and it's pretty discontinuous. Just as a topological space, it is quite nice - it's a compact one-dimensional manifold-with-boundary. – Izaak van Dongen May 09 '24 at 17:47
  • I can explain what I mean by metaphor. Consider the image $\tan^{-1}(x)$ over $\mathbb{R}\cup \infty$. There is a "homomorphism" here via $\tan^{-1} \left( \frac{x+y}{1-xy} \right) = \tan^{-1}(x) + \tan^{-1}(y)$ so the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$ with the group structure addition modulo $\pi$ should be viewed as some kind of non-discontinuous thing. It's kind of like a copy of $S^1$ (if we identify $-\frac{\pi}{2}, \frac{\pi}{2}$ except there are half as many angles here). It also seems to have a connection with $Gr(1,2)$ in the sense that... – Sidharth Ghoshal May 09 '24 at 17:55
  • ...every value on that interval is related to a line passing through the origin and $\tan^{-1}$ of the slope of the line produces an angle on the line. And so for some reason I feel like it is its own distinct thing, but I cannot articulate any property of this space that differentiates it from the standard representation of $S^1$. (Yet we also know the standard representations of S^1 and Gr(1,V) are not the same so these both CANNOT be identifies with our arctangent image space!). I am concerned I am overcomplicating the matter, and that this is just all really $S^1$ with unusual behavior. – Sidharth Ghoshal May 09 '24 at 17:56
  • @IzaakvanDongen that tangent angle sum formula would be the "nice" or "natural" structure inherited from $\mathbb{R} \cup {\infty}$ to this space that I am talking about. – Sidharth Ghoshal May 09 '24 at 18:01
  • Initially I had naively said things like $\tan^{-1}$ is a map from $\mathbb{R} \rightarrow \mathbb{S}^1$ but that obviously is NOT the right way to view it. Yes it produces angles as output, but it produces angles in the range from $-\frac{\pi}{2}, \frac{\pi}{2}$ yet that slice of angles should be viewed as a continuous structure of its OWN. with its own addition formula. I think it's also wrong to say that $\tan^{-1} : \mathbb{R} \rightarrow Gr(1,2)$ so I guess I'm just dumping word soup attempting to find the right $X$ to confidently say $\tan^{-1}: \mathbb{R} \rightarrow X$ – Sidharth Ghoshal May 09 '24 at 18:03

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Your space also has 180 as a fixed point. And there's a lovely map from the closed unit interval to your space, namely $$ x \mapsto [180*x] $$ where brackets denote "the equivalence class of".

Because your space is diffeomorphic to this well-known manifold-with-boundary (the unit interval), a topologist would probably just call it "the unit interval".

John Hughes
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  • I am curious, since the unit interval lacks a nice group structure outside of addition mod 1 (which makes it just a copy of $S^1$). Then similarly is this "arc" with an addition that wraps around simply just a slightly more complicated version of $S^1$? – Sidharth Ghoshal May 09 '24 at 17:28