Let \(w\) be a proper labelling scheme for a 10-sided polygonal region. If \(w\) is of projective type, which of the list of spaces in Theorem 77.5 can it represent? What if \(w\) is of torus type?
Solution
Goal: Determine all topological surfaces from Theorem 77.5 represented by a proper labeling scheme \(w\) on a 10-sided polygon, in both the projective and torus cases.
1. General classification principles (§77):Proof:
1.1. A proper labeling scheme \(w\) on a 10-sided polygon has \(m = 5\) pairs of matched edges.
1.2. By the classification reduction algorithm (Theorem 77.5):- A scheme of projective type reduces to a standard non-orientable normal form \(c_1 c_1 c_2 c_2 \cdots c_k c_k\), representing the connected sum of \(k\) real projective planes (\(k \mathbb{P}^2\) or \(P_k\)). Each cross-cap block \(c_i c_i\) uses \(2\) edges.- A scheme of torus type reduces either to the empty word / 2-sphere (\(S^2\)) or to a standard orientable normal form \((a_1 b_1 a_1^{-1} b_1^{-1}) \cdots (a_g b_g a_g^{-1} b_g^{-1})\), representing the connected sum of \(g\) tori (\(g T^2\) or \(T_g\)). Each handle block uses \(4\) edges.
1.3. In the reduction steps, collapsing elementary adjacent inverse pairs (\(x x^{-1}\)) reduces the edge count by \(2\). Thus the standard form has \(2k \le 10\) edges (for projective type) and \(4g \le 10\) edges (for torus type).
2. Case 1: \(w\) is of projective type.Proof:
2.1. Since \(w\) is of projective type, at least one pair of edges occurs with the same orientation, so the genus is non-orientable (\(k \ge 1\)).
2.2. The number of projective planes \(k\) satisfies \(2k \le 10\), which implies \(1 \le k \le 5\).
2.3. Each value \(k \in \{1, 2, 3, 4, 5\}\) can be realized by adding \(5 - k\) pairs of adjacent cancelling edges \(x x^{-1}\) to the standard \(2k\)-sided form.
2.4. Thus, \(w\) can represent:- \(k = 1\): \(\mathbb{P}^2\) (the real projective plane),- \(k = 2\): \(2\mathbb{P}^2 = \mathbb{P}^2 \# \mathbb{P}^2\) (the Klein bottle),- \(k = 3\): \(3\mathbb{P}^2\),- \(k = 4\): \(4\mathbb{P}^2\),- \(k = 5\): \(5\mathbb{P}^2\).
3. Case 2: \(w\) is of torus type.Proof:
3.1. Since \(w\) is of torus type, every letter appears once with exponent \(+1\) and once with exponent \(-1\), so the resulting surface is orientable.
3.2. The number of handles \(g\) satisfies \(4g \le 10\), which implies \(g \in \{0, 1, 2\}\).
3.3. Each value \(g \in \{0, 1, 2\}\) is realized:- \(g = 0\): \(S^2\) (the 2-sphere, when all 5 pairs cancel as trivial bubbles),- \(g = 1\): \(T^2\) (the torus, using 4 edges plus 3 cancelled pairs),- \(g = 2\): \(T^2 \# T^2\) (the double torus, using 8 edges plus 1 cancelled pair).
3.4. Thus, \(w\) can represent:- \(S^2\) (the 2-sphere),- \(T^2\) (the torus),- \(T^2 \# T^2\) (the double torus).
4. Conclusion:- Projective type: \(\mathbb{P}^2, \, 2\mathbb{P}^2, \, 3\mathbb{P}^2, \, 4\mathbb{P}^2, \, 5\mathbb{P}^2\).- Torus type: \(S^2, \, T^2, \, T^2 \# T^2\).Q.E.D.