I want to calculate the fundamental group of a genus-2 surface, i.e. a double torus.

Using Van-Kampen I obtain ( with the notation generators- relations) $$\Pi_1(X,p) = < \alpha, \beta, \alpha_1, \beta_1 | \alpha \beta \alpha^{-1}\beta^{-1}, \ \alpha_1 \beta_1 \alpha_1^{-1}\beta_1^{-1}, \ \beta \beta_1^{-1} >$$

Is it correct ?


1 Answer 1


This is incorrect.

Let $[x, y] = xyx^{-1}y^{-1}$ be the commutator of $x$ and $y$.

A genus $2$ surface can be constructed by starting with a wedge sum of $4$ circles $\{a_1, b_1, a_2, b_2\}$ and attaching a $2$-cell along $[a_1, b_1] [a_2, b_2]$. The fundamental group is then $$ \langle a_1, b_1, a_2, b_2 \mid [a_1, b_1] [a_2, b_2]\rangle. $$

The abelianization of this group is $\mathbb Z^4$. The abelianization of the group you have is $\mathbb Z^3$. Thus, they are different.

  • 1
    $\begingroup$ For a newbie in this area could you please explain what this notation of <{generators} | {some elements?}> means? $\endgroup$
    – gen
    Commented Oct 15, 2018 at 20:34
  • $\begingroup$ Am I right to say that since a genus-2 surface can be glued from an octagon (see, e.g. youtube.com/watch?v=G1yyfPShgqw) and so reading from its boundary, $[a_1,b_1][a_2,b_2]$ can be retracted to a point, and thus the fundamental group as given above? $\endgroup$
    – wilsonw
    Commented Feb 25, 2019 at 9:31

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