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

Use two sheet points $v_0,v_1$. Attach a copy of ${\mathbb{RP}}^2$ at each, each mapping identically onto the projective-plane summand. Put an oriented $x$ edge from $v_0$ to $v_1$ and another from $v_1$ to $v_0$, each mapping once along the cut-open circle. This is a connected two-sheeted cover.

Its underlying two-edge circle contributes one infinite cyclic free factor, and its two attached projective planes contribute two factors of order two. Collapsing one of the connecting edges, or applying Seifert-van Kampen theorem, givesChoose the first $x$ edge as a tree path to $v_1$. The circle loop projects to $x^2$, and the two projective-plane generators project to $y$ and $xy{x}^{-1}$. By injectivity from the preceding proof,The order-two relations come from the respective projective planes; there are no additional relations between these free factors.