Consider the covering map \(p \times p: \mathbb{R} \times \mathbb{R} \to S^1 \times S^1\) of Example 4 of §53. Consider the path
\[ f(t) = (\cos 2\pi t, \sin 2\pi t) \times (\cos 4\pi t, \sin 4\pi t) \]
in \(S^1 \times S^1\). Sketch what \(f\) looks like when \(S^1 \times S^1\) is identified with the doughnut surface \(D\). Find a lifting \(\overline{f}\) of \(f\) to \(\mathbb{R} \times \mathbb{R}\), and sketch it.
Solution
1. Geometric description on the torus surface \(D\):Proof:
1.1. Under the standard identification of \(T^2 = S^1 \times S^1\) with the surface of a doughnut in \(\mathbb{R}^3\), the first factor parameterizes the longitudinal direction (going once around the central donut hole) and the second factor parameterizes the meridional direction (going around the circular cross-section of the tube).
1.2. As \(t\) increases from \(0\) to \(1\), the first coordinate \(p_1(f(t)) = (\cos 2\pi t, \sin 2\pi t)\) completes \(1\) full revolution around the longitudinal circle.
1.3. Concurrently, the second coordinate \(p_2(f(t)) = (\cos 4\pi t, \sin 4\pi t)\) completes \(2\) full revolutions around the meridional circle.
1.4. Thus, the loop \(f\) is a \((1, 2)\)-curve on the torus, winding once around the central hole while wrapping twice through the tube.
2. Explicit formula for the path lifting \(\overline{f}\):Proof:
2.1. The standard covering map \(p: \mathbb{R} \to S^1\) is given by \(p(x) = (\cos 2\pi x, \sin 2\pi x)\), so the product covering map is \((p \times p)(x, y) = (p(x), p(y))\).
2.2. Let \(\overline{f}: [0, 1] \to \mathbb{R}^2\) be defined by:\[\overline{f}(t) = (t, 2t) \quad \text{for } t \in [0, 1].\]
2.3. Continuity: The components \(\overline{f}_1(t) = t\) and \(\overline{f}_2(t) = 2t\) are linear, hence \(\overline{f}\) is continuous.
2.4. Lifting property: For every \(t \in [0, 1]\):\[(p \times p)(\overline{f}(t)) = (p(t), p(2t)) = (\cos 2\pi t, \sin 2\pi t) \times (\cos 4\pi t, \sin 4\pi t) = f(t).\]
2.5. Initial point: \(\overline{f}(0) = (0, 0)\).
2.6. By uniqueness of path liftings, this is the unique lifting of \(f\) starting at the origin \((0, 0) \in \mathbb{R}^2\).
2.7. (Any other lifting starting at an arbitrary integer lattice point \((n, m) \in \mathbb{Z}^2\) is given by \(\overline{f}_{(n, m)}(t) = (n + t, m + 2t)\).)
3. Geometric description in the universal covering plane \(\mathbb{R}^2\):Proof:
3.1. The lifting \(\overline{f}(t) = (t, 2t)\) is the straight line segment in the Cartesian plane \(\mathbb{R}^2\) connecting the origin \((0, 0)\) to the lattice point \((1, 2)\).
3.2. The endpoint \(\overline{f}(1) = (1, 2)\) reflects the winding numbers \((1, 2) \in \pi_1(S^1 \times S^1) \cong \mathbb{Z} \times \mathbb{Z}\). Q.E.D.