## regular homotopy

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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

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An obvious question: Whitney's paper is 8 pages long, and at the time there was probably no big machine available -- have you tried looking at it? – Igor Rivin Nov 23 2011 at 18:28
Yes i have read it but maybe you misunderstood my question. In Whitney's paper, he computes the connected components of the space of immersions of the circle , not the fundamental group. – nikitas Nov 23 2011 at 18:57

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Perhaps knowing the answer will help you outline a proof. It is known by Smale's work that The space of immersed loops in the plane is homotopy equivalent to the free loop space of the unit tangent bundle of the plane. In other words, the homotopy type is that of $S^1×\mathbb{Z}$. Therefore, each component of the space of immersions has $\mathbb{Z}=\pi_1(S^1)$ as its fundamental group. Moreover, given a loop $\gamma$, as you rotate the loop you generate a based loop $\Gamma$ in the space of immersed loops. This is a generator of the fundamental group of the component containing $\gamma$. – Somnath Basu Nov 23 2011 at 19:02
Ah, ok, I did misread your question... – Igor Rivin Nov 23 2011 at 19:18
@Basu, thank you , but i have read Smale's paper but the thing is i want to show it to the classroom with elementary methods if it possible. Not just say " in his paper smale defined certain fibrations etc" It is the technical part not the presentation – nikitas Nov 23 2011 at 22:46

## 1 Answer

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

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

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