Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 21H (Algebraic Topology)

The polygon presentation of the Klein bottle, or Seifert-van Kampen theorem applied to its cell structure, givesSend $a$ to $(12)$ and $b$ to $(123)$ in $S_3$. Conjugation by $(12)$ inverts $(123)$, so this defines a surjective group homomorphism. Since $S_3$ is nonabelian, the fundamental group is nonabelian.

The genus-two surface has fundamental group $G=⟨a_1,b_1,a_2,b_2\mid [a_1,b_1][a_2,b_2]=1⟩$. Connected double covering spaces correspond to index-two subgroups. Each is the kernel of a nonzero homomorphism $G\to C_2$; conjugacy creates no identifications because these kernels are normal. The commutator relation is automatic in $C_2$, so there are $2^4-1$ choices:To obtain a surjection onto the free group $F_2=⟨x,y⟩$, send $a_1\mapsto x$, $a_2\mapsto y$ and $b_1,b_2\mapsto 1$. The defining relation again vanishes, and the images generate $F_2$.