The following article gives an intuitive explanation of de Rham cohomology. In it, and other articles (such as Wikipedia) there is mention of the one-form $d\theta$ for $\theta=atan2(ky,kx)$ giving the de Rham cohomology of the punctured plane. Essentially, it gives the quotient of space of closed k-forms modulo exact ones, and is a way of counting the number of holes.
But, for the same $d\theta$ above, the winding number is $2\pi i\oint d\theta$. Since this is closely tied to the argument principle, it contributes to the statement of the principle, giving the number of zeros minus the number of poles of a function, with multiplicity.
First of all, am I right in saying that there is therefore a correspondence between the argument principle and the de Rham cohomology of the punctured plane?
If so, how can I intuitively understand the role of 'multiplicity', 'zeros' and 'poles' in terms of lingo from de Rham cohomology? For example, if I have a function that gives me a zero with multiplicity 2, is this equivalent to saying that I have '2 holes at the same point'? What about 'poles', then? I guess I am asking how to equate the notion of 'holes' to 'zeros' and 'poles.
I understand this question might sound nonsensical, but I just have a very basic understanding of these concepts and first would like to develop an intuitive understanding.