Consider the covering map \(p: \mathbb{R} \times \mathbb{R}_+ \to \mathbb{R}^2 - 0\) of Example 6 of §53. Find liftings of the paths
\[ f(t) = (2 - t, 0), \]
\[ g(t) = ((1 + t) \cos 2\pi t, (1 + t) \sin 2\pi t), \]
\[ h(t) = f * g. \]
Sketch these paths and their liftings.
Solution
The covering map is \(p: \mathbb{R} \times \mathbb{R}_+ \to \mathbb{R}^2 \setminus \{0\}\), \(p(\theta, r) = (r \cos 2\pi \theta, r \sin 2\pi \theta)\).
1. Lifting of \(f(t) = (2 - t, 0)\) for \(t \in [0, 1]\):Proof:
1.1. For all \(t \in [0, 1]\), \(2 - t > 0\), so \(f(t)\) lies on the positive \(x\)-axis at radius \(r(t) = 2 - t\) and angle \(2\pi \theta(t) = 0 \pmod{2\pi}\).
1.2. Choosing an initial lift point in \(p^{-1}(f(0)) = p^{-1}(2, 0) = \{(n, 2) : n \in \mathbb{Z}\}\), the continuous lift starting at \((n, 2)\) is:\[\tilde{f}_n(t) = (n, 2 - t) \quad \text{for } t \in [0, 1].\]
1.3. For \(n = 0\), \(\tilde{f}_0(t) = (0, 2 - t)\), which is a straight vertical segment in \(\mathbb{R} \times \mathbb{R}_+\) running from \((0, 2)\) down to \((0, 1)\).
2. Lifting of \(g(t) = ((1 + t) \cos 2\pi t, (1 + t) \sin 2\pi t)\) for \(t \in [0, 1]\):Proof:
2.1. The path \(g(t)\) is an expanding spiral in \(\mathbb{R}^2 \setminus \{0\}\) starting at \((1, 0)\) and ending at \((2, 0)\) after winding once counterclockwise around the origin.
2.2. The radius is \(r(t) = 1 + t\) and the normalized angle is \(\theta(t) = t\).
2.3. For an initial lift point \((m, 1) \in p^{-1}(g(0))\), the continuous lift is:\[\tilde{g}_m(t) = (m + t, 1 + t) \quad \text{for } t \in [0, 1].\]
2.4. For \(m = 0\), \(\tilde{g}_0(t) = (t, 1 + t)\), which is a straight diagonal line segment in the \(\theta r\)-plane from \((0, 1)\) to \((1, 2)\).
3. Lifting of the concatenation \(h(t) = (f * g)(t)\):Proof:
3.1. The concatenated path \(h: [0, 1] \to \mathbb{R}^2 \setminus \{0\}\) is given by:\[h(t) = \begin{cases} f(2t) = (2 - 2t, 0) & \text{for } 0 \le t \le 1/2, \\ g(2t - 1) = (2t \cos 2\pi(2t-1), 2t \sin 2\pi(2t-1)) & \text{for } 1/2 \le t \le 1. \end{cases}\]
3.2. By the path-lifting property for concatenations, \(\tilde{h} = \tilde{f} * \tilde{g}\).
3.3. Starting at \((n, 2)\), the unique lift is:\[\tilde{h}_n(t) = \begin{cases} (n, 2 - 2t) & \text{for } 0 \le t \le 1/2, \\ (n + 2t - 1, 2t) & \text{for } 1/2 \le t \le 1. \end{cases}\]
3.4. For \(n = 0\), the lift \(\tilde{h}_0\) starts at \((0, 2)\), moves vertically down to \((0, 1)\) at \(t = 1/2\), and then moves along the diagonal segment to \((1, 2)\) at \(t = 1\).
4. Geometric description / sketches:- In \(\mathbb{R}^2 \setminus \{0\}\): \(f\) is a segment on the positive \(x\)-axis from \((2, 0)\) to \((1, 0)\); \(g\) is a counterclockwise spiral from \((1, 0)\) to \((2, 0)\); \(h\) traces the inward radial segment and then the outward spiral.- In the covering space \(\mathbb{R} \times \mathbb{R}_+\): \(\tilde{f}_0\) is the vertical line segment from \((0, 2)\) to \((0, 1)\); \(\tilde{g}_0\) is the straight line segment from \((0, 1)\) to \((1, 2)\); \(\tilde{h}_0\) is the polygonal path connecting \((0, 2) \to (0, 1) \to (1, 2)\). Q.E.D.