I was told that the fundamental group of the projective plane with g handles is isomorphic to $\langle c_1, \ldots, c_{2g+1} | c_1^2 \cdot \ldots \cdot c_{2g+1}^2\rangle$. How can I show it?
I can think about this topological space as of factorization of a polygon. Namely, it's a polygon with $2 + 4g$ sides: $cc a_1 b_1 a_1^{-1} b_1^{-1} \ldots a_g b_g a_g^{-1} b_g^{-1}$, which are glued in a standard way. So we have a cellular decomposition with one 2-cell, so the fundamental group of the space is $\langle c, a_1, b_1, \ldots, a_g, b_g | c^2 \cdot [a_1, b_1] \cdot \ldots \cdot [a_g, b_g]\rangle$. So it remains to show that it's isomorphic to the aforementioned one.
I didn't succeed neither with cutting that polygon and gluing parts to get another word on it's boundary (like one can do with the Klein bottle to get $a^2 b^2$ from a more usual representation with $abab^{-1}$) nor with direct algebraic arguments (e.g. to find Tietze transformations from one representation to another).