regular homotopy - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T00:04:09Z http://mathoverflow.net/feeds/question/81730 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81730/regular-homotopy regular homotopy nikitas 2011-11-23T18:21:12Z 2012-10-02T11:48:49Z <p>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 ?</p> <p>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.</p> <p>Any idea would be much appreciated</p> http://mathoverflow.net/questions/81730/regular-homotopy/108627#108627 Answer by Peter Michor for regular homotopy Peter Michor 2012-10-02T11:48:49Z 2012-10-02T11:48:49Z <p>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. <a href="http://www.mat.univie.ac.at/~michor/curves.pdf" rel="nofollow">pdf</a></p> <p>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. <a href="http://www.mat.univie.ac.at/~michor/immersions%5E0.pdf" rel="nofollow">pdf</a></p>