Past exam of the topology and Groups and Algebraic Topology course of the University of Oxford / 2009 / A / 3 / ii

Take the sphere double cover $S^2\to {\mathbb{RP}}^2$ and denote the two points above the wedge point by $v_0,v_1$. Attach one $x$ circle at each point, mapping identically onto the circle summand. At both points there is one incoming and one outgoing $x$ branch, so this is a connected two-sheeted cover of the wedge.

The sphere is simply connected, while each attached circle supplies one generator. Join the two attachment points by an arc in the sphere and collapse that contractible arc; the fundamental group is the free product of the two circle groups, hence free of rank two. A lift of $y$ joins $v_0$ to $v_1$. The two loop generators therefore project toInjectivity of the covering map identifies their subgroup with the free group on these two generators.