Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 21H (Algebraic Topology)

Take a CW complex with one vertex, loops labelled $a,b$, and one two-cell attached along $a^2{b}^3{a}^3{b}^2$. The Seifert-van Kampen theorem gives the required fundamental group. Connected double covering spaces correspond to index-two subgroups, equivalently kernels of surjective homomorphisms $G\to C_2$. The relation imposes $5a+5b=0$ modulo two, so the unique nonzero possibility sends both $a,b$ to one. Thus there is a unique connected double cover up to covering isomorphism.

Construct it with two vertices; both labelled edges switch sheets, and attach the two lifts of the two-cell. Contract one of the $a$ edges to a spanning tree. With coset representatives $1,a$, the remaining Schreier generators can be taken as $A=a^2$, $B=b{a}^{-1}$ and $C=ab$. Reading the two lifts of the attaching word, starting on the two respective sheets, gives $ABCB{A}^{2}BC$ and $ACBCACB$. ThusFor example, a letter $a$ contributes nothing on the first sheet and $A$ on the second; a letter $b$ contributes $B$ on the first and $C$ on the second. Both letters then switch sheets, which directly verifies the two relators.