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I'd like to compute the fundamental (using Seifert-van Kampen theorem) group of a genus-2 surface and others inductively made surfaces (like connected triple torus ).I know that to compute the group of the first one I have to represent it as a quotient of a plain polyhedron (it's analogous to computing the f. group of a torus by representing it as a sguare).But I don't know how to do it in a latter examples.Is it posibble to somehow do it inducitevely?

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  • $\begingroup$ They can all be computed from polygonal presentations. It's not hard to guess the pattern. $\endgroup$
    – Pedro
    Mar 26 '16 at 15:45
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When you take out a disk from a torus, then the fundamental group of the resulting punctured torus is a free group in two generators and the boundary curve represents the commutator of the two generators. (This can be seen by taking a disk out of the square and then having a homotopy equivalence to the boundary of that square. That yields a homotopy equivalence of the punctured torus to its 1-skeleton, whose fundamental group is free. The boundary curve of the punctured torus is homotopic to the boundary of the square, which runs through a, b, a^{-1}, b^{-1}.)

The surface of genus 2 is obtained from two punctured tori by identifying their boundaries. So by Seifert-van Kampen its fundamental group is obtained as an amalgamated product from two copies of the 2-generator free group by identifying their commutators.

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