From: Robert Bryant
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