Give examples to show that a simple closed curve in the torus may or may not separate the torus.
Solution
Goal: Provide explicit examples of simple closed curves (embedded circles \(S^1 \hookrightarrow T^2\)) in the torus \(T^2 = S^1 \times S^1\) that (1) do not separate \(T^2\) and (2) do separate \(T^2\).
1. Example of a non-separating simple closed curve:Proof:
1.1. Identify the torus as \(T^2 = S^1 \times S^1\), and fix a point \(y_0 \in S^1\).
1.2. Define the curve \(C_1 = S^1 \times \{y_0\}\).
1.3. The map \(\iota_1: S^1 \to T^2\) given by \(\iota_1(\theta) = (\theta, y_0)\) is an embedding, so \(C_1\) is a simple closed curve (a longitudinal circle).
1.4. The complement is:\[T^2 \setminus C_1 = S^1 \times (S^1 \setminus \{y_0\}).\]
1.5. Since \(S^1 \setminus \{y_0\} \cong (0, 1)\) is homeomorphic to an open interval, \(T^2 \setminus C_1 \cong S^1 \times (0, 1)\) is homeomorphic to an open cylinder.
1.6. The open cylinder \(S^1 \times (0, 1)\) is path-connected (connected).
1.7. Thus \(C_1\) does not separate \(T^2\).
2. Example of a separating simple closed curve:Proof:
2.1. Choose an open coordinate chart \(U \subset T^2\) homeomorphic to \(\mathbb{R}^2\).
2.2. Inside \(U\), let \(C_2\) be a small geometric circle of radius \(r > 0\) bounding a closed 2-disk \(D = \overline{B_r(\mathbf{0})} \subset U\).
2.3. \(C_2 = \partial D \subset T^2\) is an embedded simple closed curve (a contractible loop on the torus).
2.4. The complement \(T^2 \setminus C_2\) decomposes into two disjoint non-empty open sets:\[U_1 = B_r(\mathbf{0}) \quad \text{(the interior open disk)},\]\[U_2 = T^2 \setminus D \quad \text{(the exterior punctured torus)}.\]
2.5. Both \(U_1\) and \(U_2\) are non-empty and open in \(T^2 \setminus C_2\), so \(T^2 \setminus C_2 = U_1 \cup U_2\) is disconnected.
2.6. Thus \(C_2\) separates \(T^2\).
3. Conclusion:\(C_1\) is non-separating and \(C_2\) is separating on the torus \(T^2\). Q.E.D.