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I am trying to understand what is meant by the following wording on page 297 of John Lee's Introduction to Smooth Manifolds, the 2nd Edition:

"The key to constructing a potential function in Theorem 11.49 is that we can reach every point $x \in M$ by a definite path from $c$ to $x$, chosen to vary smoothly as $x$ varies. This is what fails in the case of the closed covector field $\omega$ on the punctured plane (Example 11.43): because of the hole, it is impossible to choose a smoothly varying family of paths starting at a fixed base point and reaching every point of the domain exactly once."

Theorem 11.49: If $U$ is a star-shaped open subset of $\mathbb{R}^{n}$ or $\mathbb{H}^{n}$, then every closed covector field on $U$ is exact.

In Example 11.43, the covector field is

$$ \omega = \frac{xdy - ydx}{x^{2} + y^{2}} $$

and the book demonstrates that integrating this over the unit cirlce gives $2\pi$. My understanding is (a bit informally) that path independence of a line integral is equivalent to the line integral over any loop being 0 which is the same as every loop being homotopic to a point, and this is what goes wrong. Is Lee saying there is another way to see what goes wrong? Or is this a different wording for what I just mentioned? Thanks in advance.

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Lee is pointing out exactly what step fails if you try to use the proof of Theorem 11.49 in a punctured plane.

Theorem 11.49 (the Poincare lemma) gives you a recipe for creating a potential out of any covector, if you can find that smoothly-varying way to get from your starting point to anywhere in the domain.

If you can't, the recipe isn't guaranteed to work any more. Sometimes it does (there are exact covectors in the punctured plane, of course) but you can't guarantee it.

So you look for a covector like $\omega$ that might actually care what path you take to get to a point. Check: does it matter what path you take to get to a point when you try to construct a potential? Equivalently, are there any loops with nonzero path integral? and lo and behold, there are!

(You have just found a way to measure the topology of a space, such as the number of points it has, by using analysis. Welcome to differential topology. :)

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