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By my course, all manifold of dimension 1 is isomorphic to $(0,1), (0,1],[0,1)$ or $\mathbb S^1=\{x^2+y^2=1\mid x,y\in\mathbb R\}$.

I was thinking of a curve in the plan, with a knot. (See picture) enter image description here I agree that the gluing (in pink) is possible in $\mathbb R^3$, but if we consider this curve in $\mathbb R^2$, it doesn't look possible to me, otherwise we have to cut the curve like this

enter image description here and thus, It can't work, no ? Because if it does, we could do the same with the torus, and then, the Torus in $\mathbb R^3$ would be homeomorphic to the sphere $\mathbb S^3=\{x^2+y^2+z^2=1\mid x,y,z\in\mathbb R\}$, which is not the case. Am I right ?

Q1) So if I'm right, don't we have to add those type of curve (i.e. with a knot) in the classification of manifold of dimension 1 ?

Q2) If not, why the torus is not homeomorphic to $\mathbb S^2$ in $\mathbb R^3$ ?

Rick
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  • I don't understand your reasoning, but note that the figure eight curve you drew in the first line is not a one dimensional manifold (assuming you endow it with the induced topology from $\mathbb{R}^2$). The self-intersection point doesn't have an open neighborhood that is homeomorphic to an open interval. – levap Nov 30 '15 at 09:53
  • interesting... I didn't think about the fact that it can't be a manifold... – Rick Nov 30 '15 at 09:56
  • @levap: forget the fact that the curve eight is seen as a subset of $\mathbb R^2$ (actually, it's a stupid idea). We can simply take ce circle and gluing the pole north and the pole sud. Then, it's a manifold, isn't it ? – Rick Nov 30 '15 at 10:05
  • No. Again, an open neighborhood of the glued point (with the quotient topology) is not homeomorphic to an open interval. – levap Nov 30 '15 at 10:07
  • How can I prove it ? And for the torus with 2 anses ? (I ask this because the situation looks similar). @levap – Rick Nov 30 '15 at 10:09
  • If you remove a point from an open interval, the interval is separated into two connected components. If you remove a point from a small open neighborhood of the glued point, it will be separated into four connected components. – levap Nov 30 '15 at 10:16

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