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



regular homotopy
See theorem 2.10 (with elementary proof) for the case of rotation idex $\ne 0$ of the paper:
Peter W. Michor; David Mumford: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8 (2006), 1-48. [pdf](http://www.mat.univie.ac.at/~michor/curves.pdf)


For rotation index $=0$ (with a somewhat surprising answer) see the paper: 
Hiroki Kodama, Peter W. Michor: The homotopy type of the space of degree 0 immersed curves. Revista Matemática Complutense 19 (2006), no. 1, 227-234. [pdf](http://www.mat.univie.ac.at/~michor/immersions%5E0.pdf)


