From: Robert Bryant <bryant@math.duke.edu>
Subject: Re: immersions of circles into the plane
Date: Thu, 29 Apr 1999 10:29:27 -0400
Newsgroups: sci.math.research
Keywords: Whitney's Theorem

v g pestov wrote:
> 
> Hello,
> 
> has anyone in the past classified all immersions of the circle
> $S^1$ into the Euclidean plane $R^2$ up to a regular homotopy?
> 
> I cannot really believe a result that basic cannot be found  
> somewhere as a textbook exercise!  ;)*

	Fortunately, you don't have to believe such a thing because
this does appear as an exercise in a 'textbook'.  The result
you want is known as Whitney's Theorem.  It appears as an
exercise in M. Gromov's "Partial Differential Relations" (pg. 14).
Oddly, many people don't think of this as a textbook. ;)

	Of course there is a far more general theorem of which
this is a special case:  The Hirsch-Smale Immersion Theorem
(see PDR, section 1.1.3).

	Yours,

	Robert Bryant