Source: Guillemin-Pollack Exercise 4.8.2.

Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 - \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 - \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where $W(\gamma, 0)$ is the winding number of $\gamma$ with respect to the origin. $W(\gamma, 0)$ is defined just like $W_2(\gamma, 0)$, but using degree rather than degree mod $2$; that is, $W(\gamma, 0) = \text{deg}(\gamma/|\gamma|)$. In particular, conclude that$$W(\gamma, 0) = {1\over{2\pi}}\oint_\gamma d\,\text{arg}.$$

My solution:

Consider the mapping $f: \gamma \mapsto S^1$ defined by $f(x) = x/|x|$. Then we can consider the pullback of $\omega$ under this mapping to get$$\oint_\gamma f^*\omega = (\deg f)\int_{S^1} \omega = W(\gamma, 0) \int_{S^1} \omega = 2\pi\,{W(\gamma, 0)}.$$Therefore, if we can show that $f: \mathbb{R}^2 - \{0\} \mapsto S^1$ is homotopic to the identity mapping, we will be able to show the desired result. Consider the homotopy $H(x, t) = x/(t|x| + (1-t))$. This is the desired homotopy, so the result holds because$$\oint_\gamma f^*\omega = \oint_\gamma \iota^* \omega = \oint_\gamma \omega = \oint_\gamma d\,\text{arg}.$$My question is, is what I have correct? I feel rather unsure with the material. For starters, is $f: \gamma \mapsto S^1$ defined by $f(x) = x/|x|$ the right notation/even valid? Does anyone more experienced have any intuition they could possibly help?


For the most part, the proof is correct. The definition of the map $f$ makes perfect sense. Pullback under $f$ makes sense enough (although at this point you're really pulling back under $\iota\circ f$ where $\iota$ is the inclusion of $S^1$ into $\mathbb{R}^2$).

But this part is both wrong and unnecessary:

$$ W(\gamma, 0) \int_{S^1} \omega = 2\pi\,{W(\gamma, 0)}$$

There's no reason for $\int_{S^1} \omega$ to be equal to $2\pi$. It can be any real number whatsoever; after all, one can multiply $\omega$ by a constant.

Similarly, this step is both wrong and unnecessary:

$$ \oint_\gamma \omega = \oint_\gamma d\,\text{arg}.$$

Eliminating the wrong steps, you end up with a chain of equalities $$W(\gamma, 0) \int_{S^1} \omega = \cdots = \oint_\gamma \omega$$ which is what was required.


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