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I have to draw a picture of the base of the first homology group over $\mathbb{Z}$ of $C$, where $C$ is a compact Riemann surface of positive genus.
How can i draw it? Is there some free programm wich can do it? Or do you know some site where i can find the picture?

dario
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See for example Figure $3.1$ here, for the case of genus $g=3$. Can you generalize this for arbitrary genus $g\ge 1$ ? In general we have $H_1(M,\mathbb{Z})\simeq \mathbb{Z}^{2g}$, for a compact Riemannian surface $M$ of genus $g$.

The homology is generated by the homology classes of loops going around each of the $g$ holes of $M$ together with the homology classes of loops transverse to these, going around each of the handles. So we have $2g$ loops.

Dietrich Burde
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  • yes but the problem is to draw the direct sum of g torus to rappresent my Riwmann surface. How can i picture it? i don't if there is a standard way to rappresent the direc sum of g torus – dario May 12 '15 at 14:10
  • The connected sum of $g$ tori, see Figure $3.1$. – Dietrich Burde May 12 '15 at 14:12