If we define an equivalence relation in $\mathbb{S}^1\times [0,1]$ where $(x,0)$ is identified with $(-x,1)$, which space do we obtain?
My intuition says that it may be a torus, but I am not sure.
2026-04-03 06:49:58.1775198998
Cylinder with edges identified
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That's right, and you can prove it rigorously by showing that the given gluing function $f(x,1)=(-x,0)$ is isotopic to the gluing function $g(x,1)=(x,0)$ which more obviously has quotient homeomorphic to a torus.
In general, when gluing together two boundary components of a manifold to get a quotient space, isotopic gluing maps give homeomorphic quotient manifolds.