How to compute fundamental groups of closed surfaces without using Van-Kampen theorem?

Question

Denote $X=mP^2$ the sphere glued with $m$ Mobius bands. It has a polygon representation $a_1a_1...a_ma_m$, i.e. it's a quotient by a $2m$-sides polygon $P$. Let $o$ be the center point of $P$, $x_0$ a vertex of $P$.

Denote $X_1=P\backslash\{o\}$, then $X_1$ is homotopy to $\partial P$, a bouquet of $m$-circles. So $$ \pi_1(X_1,x_0)=<a_1,...,a_m>, $$

the free group generated by loops $a_1,...,a_m$.

Consider the inclusion map $$ f:\pi_1(P\backslash\{o\},x_0)\to \pi_1(P,x_0), $$ it's a surjective homomorphism. $a_1^2a_2^2...a_m^2\in \pi_1(P\backslash\{o\},x_0)$ is mapped to the trivial elment of $\pi_1(P,x_0)$.

Denote $[a_1^2...a_m^2]$ the normal subgroup generated by $a_1^2...a_m^2$, we know that $[a_1^2...a_m^2]\subset Ker f$. How to prove that $$ Ker f=[a_1^2...a_m^2] $$ without using the conclusion of Van-Kampen theorem? In this simple case, I prefer a direct proof.

Answer 1

As Ryan Budney points out, the only way to not use the ideas behind the Van Kampen theorem is to covering space theory. In the case of surfaces, almost all of them have rather famous contractible universal covers: $\mathbb R^2$ in the case of a torus and Klein bottle, and the hyperbolic plane for surfaces of higher genus. Ironically, dealing with our remaining surface -- showing that the 2-sphere is simply connected -- seems to need a Van Kampen style argument (or, as has been rightly pointed out in the comments, some approximation theorem): not hard, but curiously different.

Answer 2

Any presentation of a given space as a CW-complex immediately gives rise to a presentation of the fundamental groupoid, and hence also the fundamental group.

Specifically, given such a presentation as a CW-complex C, the objects of the fundamental groupoid G are precisely the vertices of C. Each 1-cell of C yields a generating 1-morphism of G. Finally, each 2-cell of C yields a relation for G, induced by the boundary attaching map: the boundary traverses finitely many 1-cells, and each such 1-cell contributes the corresponding generating morphism or its inverse, depending on whether we traverse it in the original direction (given by the structure of a CW-complex) or its inverse.

Another approach is to observe that the given polygon can be considered as a simplicial set, with vertices o and x_0, 1-simplices a_1, …, a_n, and b_1, c_1, b_2, c_2, …, b_n, c_n (radial spokes that connect o and x_0), and 2n different 2-simplices, with the (2i-1)st and 2ith 2-simplices having edges b_i, c_i, a_i respectively c_i, b_{i+1}, a_i. Now one can read off the fundamental groupoid in exactly the same manner as before.

Once a system of generators and relations for the fundamental groupoid is written down, it can be easily converted to a system of generators and relations for the fundamental group.