A hexagon with the identifications \(a+b+c-a-b-c\)
Solution
Goal: Identify the closed 2-dimensional surface \(X\) represented by a regular hexagon whose perimeter edges are identified according to the boundary word \(w = a b c a^{-1} b^{-1} c^{-1}\), compute its Euler characteristic \(\chi(X)\), classify the surface, and determine its fundamental group \(\pi_1(X)\).
1. Cellular structure of the polygonal quotient \(X\).
1.1. 2-cells: There is 1 two-cell \(e^2\) (the hexagon interior).
1.2. 1-cells: There are 3 one-cells, corresponding to the edge labels \(a, b, c\).
1.3. 0-cells: Determine the vertex identification classes from the boundary word \(a b c a^{-1} b^{-1} c^{-1}\).- Label the 6 vertices of the hexagon sequentially around the perimeter as \(v_0, v_1, v_2, v_3, v_4, v_5\) so that the directed edges are:- Edge 1 (\(a\)): \(v_0 \to v_1\),- Edge 2 (\(b\)): \(v_1 \to v_2\),- Edge 3 (\(c\)): \(v_2 \to v_3\),- Edge 4 (\(a^{-1}\)): \(v_3 \to v_4\) (so the second \(a\)-edge is directed \(v_4 \to v_3\)),- Edge 5 (\(b^{-1}\)): \(v_4 \to v_5\) (so the second \(b\)-edge is directed \(v_5 \to v_4\)),- Edge 6 (\(c^{-1}\)): \(v_5 \to v_0\) (so the second \(c\)-edge is directed \(v_0 \to v_5\)).- Vertex identifications:- The two \(a\)-edges are \(v_0 \to v_1\) and \(v_4 \to v_3\), so \(v_0 \sim v_4\) and \(v_1 \sim v_3\).- The two \(b\)-edges are \(v_1 \to v_2\) and \(v_5 \to v_4\), so \(v_1 \sim v_5\) and \(v_2 \sim v_4\).- The two \(c\)-edges are \(v_2 \to v_3\) and \(v_0 \to v_5\), so \(v_2 \sim v_0\) and \(v_3 \sim v_5\).- Chaining these identifications gives two equivalence classes:- Vertex Class 1: \(\{v_0, v_2, v_4\}\) (since \(v_0 \sim v_4\), \(v_4 \sim v_2\), \(v_2 \sim v_0\)),- Vertex Class 2: \(\{v_1, v_3, v_5\}\) (since \(v_1 \sim v_3\), \(v_3 \sim v_5\), \(v_5 \sim v_1\)).
1.4. Each edge label appears twice, once in each orientation, so the start and end vertices of the two copies of each edge are identified pairwise; tracing these identifications gives the two classes \(\{v_0, v_2, v_4\}\) and \(\{v_1, v_3, v_5\}\).
2. Compute the Euler characteristic \(\chi(X)\) and classify the surface.
2.1. Let \(V = 2\) (number of 0-cells), \(E = 3\) (number of 1-cells \(a, b, c\)), and \(F = 1\) (the single 2-cell).
2.2. The Euler characteristic is:\[\chi(X) = V - E + F = 2 - 3 + 1 = 0.\]
2.3. Orientability: in the boundary word \(w = a b c a^{-1} b^{-1} c^{-1}\), each letter \(a, b, c\) appears twice with opposite exponents (\(+1\) and \(-1\)), so the surface is orientable.
2.4. A closed connected orientable surface of genus \(g\) has \(\chi = 2 - 2g\); solving \(2 - 2g = 0\) gives \(g = 1\).
2.5. The Euler–Poincaré formula gives \(\chi = V - E + F = 0\), and the classification of closed orientable surfaces identifies the unique such surface as the torus \(T^2\).
3. Fundamental group \(\pi_1(X)\).
3.1. The 1-skeleton has \(V = 2\) vertices and \(E = 3\) edges, so a maximal tree \(T\) has \(V - 1 = 1\) edge; take \(T\) to be the edge \(a\).
3.2. Collapsing \(T\) leaves \(E - (V - 1) = 3 - 1 = 2\) generators, namely \(b\) and \(c\), in the quotient 1-skeleton \(X^1 / T \cong S^1 \vee S^1\).
3.3. Setting \(a = 1\) in the boundary word reduces it to \(b c b^{-1} c^{-1}\), so the relation from the 2-cell is \(b c b^{-1} c^{-1} = 1\), i.e. \(bc = cb\).
3.4. Thus \(\pi_1(X) \cong \langle b, c \mid b c b^{-1} c^{-1} = 1 \rangle \cong \mathbb{Z}^2\).
3.5. The cellular Seifert–van Kampen theorem gives \(\pi_1(X) \cong \pi_1(X^1)/\langle\langle [\phi] \rangle\rangle\); collapsing the maximal tree \(T\) identifies \(\pi_1(X^1)\) with the free group on \(b, c\), and the attaching word \(b c b^{-1} c^{-1}\) imposes the single relation \(bc = cb\), yielding \(\mathbb{Z}^2\).
4. Q.E.D.
4.1. Steps <1>1–<1>3 show that \(X\) is the torus: \(\chi(X) = 0\) and \(\pi_1(X) \cong \mathbb{Z}^2\).