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In the process of trying to find continuous parametric surface equations for the double torus and for a pair of pants, I believe that the problem is unsolvable for some topological reason.

I have found a few arguments for why you can't do it using various approaches, in which it would force a discontinuity, but I would like a solid proof of this.

The main reason this is important is because it would be quite useful to have a continuous parametrization for such a surface.

Alternatively, might I be wrong and could there actually exist continuous parametrizations for such surfaces?

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    $\begingroup$ I was wondering if you could define your terms more precisely. By "double torus" I'm guessing you mean a genus two (2-holed) surface. But what exactly do you mean by "continuous parameterization"? There is, in fact, a continuous map from the disk to a genus two surface. $\endgroup$
    – Donu Arapura
    Commented Aug 28, 2011 at 18:56
  • $\begingroup$ Instead of "to a genus..." I should have said "onto a ..." in my last sentence. $\endgroup$
    – Donu Arapura
    Commented Aug 28, 2011 at 18:58
  • $\begingroup$ On the other hand, there is no bijective continuous function from a disk to either a pair of pants or a genus-two surface. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Aug 28, 2011 at 19:06
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    $\begingroup$ (By the way, math.stackexchange.com might have been a better place to ask this question) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Aug 28, 2011 at 19:12
  • $\begingroup$ There is a bijective parametrization of the pair of pants by a disc with two holes cut out. I agree with Mariano that your question is not quite appropriate for this web site. Please see the FAQ for a list of sites where you may find more complete answers. $\endgroup$
    – S. Carnahan ♦
    Commented Aug 29, 2011 at 2:33

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