What familiar space is the quotient \(\Delta\)-complex of a 2 simplex \([\nu_0, \nu_1, \nu_2]\) obtained by identifying the edges \([\nu_0, \nu_1]\) and \([\nu_1, \nu_2]\), preserving the ordering of vertices?
Solution
Let \[ a=[\nu_0,\nu_1]\sim[\nu_1,\nu_2] \] with the identification preserving the displayed vertex order, and let \[ b=[\nu_0,\nu_2] \] be the remaining boundary edge.
1. All three vertices become a single vertex in the quotient.
The edge identification sends \[ \nu_0\mapsto\nu_1, \qquad \nu_1\mapsto\nu_2. \] Hence \[ \nu_0\sim\nu_1\sim\nu_2. \]
2. The quotient has one \(2\)-cell, one interior edge class \(a\), and one boundary edge \(b\).The attaching word of the \(2\)-cell is\[ a^2b^{-1} \]up to reversing all orientations.
Orient the triangle by \([\nu_0,\nu_1,\nu_2]\). Its oriented boundary is \[ [\nu_0,\nu_1]+[\nu_1,\nu_2]-[\nu_0,\nu_2]. \] The first two edges both map to \(a\) with the same orientation, while the third maps to \(b\). Thus the quotient polygon has boundary word \(aab^{-1}\).
3. This is the standard polygon model of the Möbius band.
A Möbius band may be obtained from a rectangle by identifying one pair of opposite sides with reversed transverse orientation. Cutting that rectangle along a diagonal gives a triangle whose two successive sides are identified in the same directed order; the third side remains the boundary circle. Equivalently, the core circle is represented by \(a\) and the boundary circle represents \(a^2\), exactly the relation \[ b=a^2 \] encoded by 2.
4. Therefore the familiar quotient space is\[ \boxed{\text{the Möbius band}.} \]
This is 3.