This is an exercise in Hatcher's Algebraic topology:
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 the edges?
The equivalence relation is thus generated by $tv_0 +(1-t)v_1 \sim tv_1+(1-t)v_2$. It is no problem to calculate the homology groups which are the same as for the circle $S^1$. The quotient could thus be homotopy equivalent to $S^1$. Nevertheless, I don't SEE this familiar space (or, more precisely, a homeomorphic copy in $\mathbb R^3$).