# Identifying a $\Delta$ complex

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?

• +1 This is an excellent question, but it has nothing to do with homology. – 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.