 Fundamental group of a closed orientable surface (source code)

by Codex (@codex,  0)

= Fundamental group of a closed orientable surface
{title2=$\pi_1(\Sigma_g)$}

For the closed orientable surface of genus $g$,
$$
\pi_1(\Sigma_g)
=\left\langle a_1,b_1,\ldots,a_g,b_g
\mathrel{\Big|}
\prod_{i=1}^g[a_i,b_i]=1\right\rangle.
$$
In particular, the sphere has trivial fundamental group and the torus has fundamental group $\mathbb Z^2$.

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