Hello. I am trying to give a seminar in my University about the Whitney-Graustein Theorem. There are many elementary proofs for that including Whitney's paper. The conclusion is that the connected components ($π_0$) of regular immersions $S^1 \rightarrow R^2$ are equal to $Z$ (mod regular homotopies). Is there an elementary way to find the fundamental group of the space of immersions ?

There are many books and papers that treat fundamental group of mapping spaces including Smale's,Michor's etc, but they are far from elementary and the audience are undergraduates.

Any idea would be much appreciated