I am doing some self study and am having trouble with the following. I want to say the answer is a cone, but I do not think that this is correct. Help will be apreciated.

What familiar space is the quotient $\Delta$ complex of a 2 simplex $[v_0 , v_1 , v_2 ]$ obtained by identifying the edges $[v_0 , v_1 ]$ and $[v_1 , v_2 ]$, preserving the ordering of vertices?

  • $\begingroup$ +1 This is an excellent question, but it has nothing to do with homology. $\endgroup$ – Georges Elencwajg Mar 21 '13 at 18:08

You get a cone if you identify the edges $[v_0, v_1]$ and $[v_1, v_2]$ reversing the ordering of the vertices (because you identify $v_0$ with $v_2$ and $v_1$ with $v_1$ when you do this).

To preserve the orientation, you need to make a "twist" to identify $v_0$ with $v_1$ and $v_1$ with $v_2$. This gives a Möbius band.

You can see more details (and pictures) here.

  • $\begingroup$ A Möbius band from a triangle: amazing! $\endgroup$ – Georges Elencwajg Mar 21 '13 at 18:06
  • $\begingroup$ It is strange for me to convince a Möbius band can be get from a triangle, simply because if I have a Möbius band and then, let a vertical line retracts to a point, WHY CAN'T I twist it back at that point? $\endgroup$ – Hugo Nov 12 '18 at 6:37

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