TITLE: Identifying a $\Delta$ complex
QUESTION [11 upvotes]: 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?

REPLY [10 votes]: 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.