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The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation $$ \langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle. $$ The proof I know ...

A basic consequence of the Seifert–van Kampen theorem is the following. Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-...