2

There are often questions in differential geometry asking if a certain manifold (say a circle) has a polynomial parametrization. Are there topological obstructions to existence of such parametrization? More precisely, if we have a polynomial map $f:U\rightarrow R^m$ (and suppose $U$ is the interior of a cube in $\mathbb{R}^n$), then

$\bullet$ can the image of $f$ have non-trivial topology? I'm guessing the answer is yes but can't think of an example.

Paulo Henrique
  • 1,797
adrido
  • 2,283
  • Every compact manifold is the continuous image of a cube of the same dimension, and every such manifold embeds into some $R^m$, so you can get lots of topology in the image of such a map! – Mariano Suárez-Álvarez Jan 19 '14 at 04:12
  • @ Mariano Suárez-Alvarez: I'm asking for $f$ to be a polynomial. does that make a difference? – adrido Jan 19 '14 at 04:17
  • Then you should say so earlier. The first reference to polynomial maps in what you wrote is at the end (it is the last word, in fact) – Mariano Suárez-Álvarez Jan 19 '14 at 04:18
  • The body of questions should be independent of the titlees, really. I in fact thought the title referred to the second bullet point in your question, reflecting the fact that it appears to be the part which interests you more. – Mariano Suárez-Álvarez Jan 19 '14 at 04:23
  • fair enough! thank you for pointing this out. – adrido Jan 19 '14 at 04:25

0 Answers0