Complex integration and the homotopy version of Cauchy’s theorem

$\textcolor{black}{\textbf{Theorem 1.19}}$

$Page \mathit{21}, Schlag: \textcolor{purple}{\textit{A course in complex analysis and riemann surface.}}$

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$\textcolor{purple}{\textbf{[Statement]}}$

$\textcolor{darkblue}{\textit{Setup}:}$
$$ \gamma_0, \gamma_1: [0,1] \to \Omega $$
are $\textcolor{brown}{C^1}$ $\textbf{curves}$, $\textcolor{brown}{C^1}-\textbf{homotopic}$ (with endpoints fixed),
in the fixed endpoint case:
$$\gamma_0(0)=\gamma_1(0), \gamma_0(1)=\gamma_1(1);$$
or in the closed case:
$$\gamma_0(1)=\gamma_1(0), \gamma_0(1)=\gamma_1(0).$$

$\textcolor{darkblue}{\textit{Output}:}$
$$ \int_{\gamma_0} f(z)dz = \int_{\gamma_1} f(z)dz, \forall f \in \mathcal{H}(\Omega). $$
$\textit{In particular}$, if $\gamma$ is closed in $\Omega$ which is homotopic to a point, then
$$\oint_{\gamma} f(z)dz = 0.$$
This is the case if $\gamma$ is the boundary of a subregion $\Omega_1 \subset \Omega$ such that
$\overline{\Omega_1}$ is $\textbf{diffeomorphic} (i.e. \textbf{bi-differentiable})$ to a closed disk.

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$\textcolor{purple}{\textit{1. Topological View}}$

We first introduce some embedding and approximation theorems, and then follows the proof. We refer to GTM222 of Lee and GTM125 of BG.

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$\textbf{[Preparations]}$

$\textcolor{darkgreen}{\textit{1. Whitney Approximation Theorem}}$

Let $M$ be a $\textcolor{brown}{\textit{smooth manifold}}$ and let $F: M \to \mathbb{R}^k$ be a $\textcolor{darkblue}{\textit{continuous}}$ function. Given any positive $\textcolor{darkblue}{\textit{continuous}}$ function $\delta: M \to \mathbb{R}$, there exists a $\textcolor{brown}{\textit{smooth}}$ function $\tilde{F}: M \to \mathbb{R}^k$ that is $\delta-$close to $F$.

$\textcolor{darkgreen}{\textit{2. Whitney Embedding Theorem}}$

$\textcolor{darkgreen}{\textit{3. Whitney Approximation Theorem on Manifolds}}$

Let $N, M$ be $\textcolor{brown}{\textit{smooth manifolds}}$, and let $F: N \to M$ be a $\textcolor{darkblue}{\textit{continuous}}$ map. Then $F$ is homotopic to a $\textcolor{brown}{\textit{smooth}}$ map $\tilde{F}: N \to M$.

$\textcolor{darkgreen}{\textit{4. Corollary}}$

If $F,G: M\to N$ are homotopic smooth maps, then they are $\textcolor{brown}{\textit{smoothly homotopic.}}$ If $F$ is homotopic to $G$ relative to some closed subset $A \subset M$, then they are smoothly homotopic relative to $A$.

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$\textbf{[Proof]}$

By virtue of these properties of homotopy, one can apply Stoke’s Theorem which demands both $M$, the oriented $n-$dimensional manifold with boundary, and $\omega$, the compactly supported $(n-1)-$form on $M$, are smooth.

$\textcolor{darkgreen}{\textit{Definition: }}$

$\textbf{Exact Differential Forms}$: $\exists f\in C^{\infty}(M)$ such that $\omega = df$.
$\textbf{Closed Differential Forms}$: $d\omega = 0$.

$\textcolor{darkgreen}{\textit{Local Exactness of Closed covector field}}$
$\omega$: a closed covector field on a smooth manifold $M$. Then every $p \in M$ has a neighborhood on which $\omega$ is exact.

Now follows naturally the proof in page 21.

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$\textcolor{purple}{\textit{2. Analytical View}}$