Let \(f\) be a loop in \(\mathbb{R}^2 - a\); let \(g(s) = [f(s) - a]/\norm{f(s) - a}\). The map \(g\) induces, via the standard quotient map \(p: I \to S^1\), a continuous map \(h: S^1 \to S^1\). Show that \(n(f, a)\) equals the degree of \(h\), as defined in E-UU8CC.
Solution
1. Lifting and definition of the winding number:
1.1. Consider the standard universal covering map \(e: \mathbb{R} \to S^1\) given by \(e(t) = (\cos 2\pi t, \sin 2\pi t) = e^{2\pi i t}\).The normalized loop \(g: [0, 1] \to S^1\) is defined by:\[ g(s) = \frac{f(s) - a}{\|f(s) - a\|}. \]
definition of radial projection onto \(S^1\).
1.2. By the Path Lifting Property for covering spaces, given any choice of \(\tilde{g}(0) = t_0 \in e^{-1}(g(0))\), there exists a unique continuous lift \(\tilde{g}: [0, 1] \to \mathbb{R}\) such that \(e \circ \tilde{g} = g\).
Path Lifting Theorem for covering spaces.
1.3. By definition, the winding number of the loop \(f\) around \(a\) is the integer change:\[ n(f, a) = \tilde{g}(1) - \tilde{g}(0). \]
standard definition of winding number via angular lifting.
2. Lifting and definition of the degree of \(h: S^1 \to S^1\):
2.1. The quotient map \(p: [0, 1] \to S^1\) is \(p(s) = e(s)\).The continuous map \(h: S^1 \to S^1\) satisfies \(h(p(s)) = g(s)\) for all \(s \in [0, 1]\).
definition of the induced map \(h\).
2.2. The standard generator of \(\pi_1(S^1, 1)\) is the path homotopy class of \(\gamma(s) = e(s)\).The induced homomorphism \(h_*: \pi_1(S^1, 1) \to \pi_1(S^1, h(1))\) maps \([\gamma]\) to \([h \circ \gamma] = [g]\).
functoriality of the fundamental group.
2.3. Under the canonical isomorphism \(\pi_1(S^1) \cong \mathbb{Z}\) that identifies a loop’s homotopy class with the difference of the endpoints of its lift in \(\mathbb{R}\):The loop \(g = h \circ \gamma\) lifts to \(\tilde{g}\), so the degree of \(h\) is:\[ \deg(h) = \tilde{g}(1) - \tilde{g}(0). \]
definition of the degree of a self-map of \(S^1\) via lifting.
3. Conclusion:Both definitions coincide with \(\tilde{g}(1) - \tilde{g}(0)\), so \(n(f, a) = \deg(h)\). Q.E.D.
1 and 2.