Past exam of the topology and Groups and Algebraic Topology course of the University of Oxford / 2007 / A / 4 / ii

Write the wedge's fundamental group as the free group on $a,b$. Use the index-two subgroupswhere each map has target $\mathbb{Z}/2$. Explicitly, the first cover has two vertices, two directed lifts of $a$ exchanging the vertices, and a $b$-loop at each vertex. The second interchanges the roles of $a,b$. Each vertex has exactly one incoming and one outgoing lift of each labelled edge, so these are genuine two-sheeted graph coverings. They are connected, and a chosen vertex gives a based cover.

The subgroups differ because $a$ belongs to $H^{\prime }$ but not $H$, while $b$ belongs to $H$ but not $H^{\prime }$. Hence the covers have the same degree and are inequivalent.