What space is indicated by each of the following labelling schemes for a collection of four triangular regions?
\(abc\), \(dae\), \(bef\), \(cdf\)
\(abc\), \(cba\), \(def\), \(dfe^{-1}\)
Solution
Goal: Identify the topological surfaces represented by each of the two triangular labeling schemes.
1. Part (a): Scheme \(abc, dae, bef, cdf\).The space is the Klein bottle (\(K^2 \cong \#^2 \mathbb{R}P^2\)).Proof:
1.1. Faces and Edges: There are \(F = 4\) triangular faces and \(E = 6\) edge identification classes (\(a, b, c, d, e, f\)).
1.2. Vertex counting:- Label the vertices of \(T_1(abc)\) as \(v_1 \xrightarrow{a} v_2 \xrightarrow{b} v_3 \xrightarrow{c} v_1\).- Trace identifications:- Class 1: \(\{v_1, w_2, z_2, w_1, u_3, z_3\}\).- Class 2: \(\{v_2, w_3, u_2, v_3, z_1, u_1\}\).- Thus there are exactly \(V = 2\) vertex equivalence classes.
1.3. Euler characteristic:\[\chi = V - E + F = 2 - 6 + 4 = 0.\]
1.4. Orientability: The edge \(a\) appears with identical cyclic orientation in both \(T_1(abc)\) and \(T_2(dae)\). Gluing along \(a\) creates a non-orientable Möbius band region.
1.5. Classification: The unique closed, connected non-orientable surface with Euler characteristic \(\chi = 0\) is the Klein bottle \(K^2\).
2. Part (b): Scheme \(abc, cba, def, dfe^{-1}\).The space is the disjoint union of the 2-sphere and the real projective plane: \(S^2 \sqcup \mathbb{R}P^2\).Proof:
2.1. Decomposition into disjoint components: The triangles \(T_1(abc)\) and \(T_2(cba)\) only involve edges \(\{a, b, c\}\), while \(T_3(def)\) and \(T_4(dfe^{-1})\) only involve edges \(\{d, e, f\}\). Since no edges or vertices are shared between the two pairs, the resulting space is the topological disjoint union \(X = X_1 \sqcup X_2\).
2.2. Component \(X_1 = T_1 \cup T_2\):- \(T_1\) has boundary word \(abc\) and \(T_2\) has boundary word \(cba = (abc)^{-1}\).- Identifying the three boundary edges of \(T_1\) with the oppositely oriented boundary edges of \(T_2\) glues two closed 2-disks along their boundary circles via an orientation-reversing homeomorphism, yielding the 2-sphere \(S^2\).
2.3. Component \(X_2 = T_3 \cup T_4\):- Gluing \(T_3(def)\) and \(T_4(dfe^{-1})\) along the shared edges \(d\) and \(f\) produces a 2-gon with remaining boundary edges identified as \(e e\) (or \(e \sim e^{-1}\)).- A 2-gon with boundary scheme \(e^2\) is the standard polygonal presentation of the real projective plane \(\mathbb{R}P^2\).
2.4. Conclusion: \(X \cong S^2 \sqcup \mathbb{R}P^2\). Q.E.D.