 Solution

ID: 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

RP^{2}

at each, each mapping identically onto the projective-plane summand. Put an oriented

x

edge from

v_{0}

v_{1}

and another from

v_{1}

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, gives

π_{1} (X, v_{0}) ≅ Z * C_{2} * C_{2} .

Choose 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

x y x^{- 1}

. By injectivity from the preceding proof,

⟨ x^{2}, y, x y x^{- 1} ⟩ ≅ Z * C_{2} * C_{2} .

The order-two relations come from the respective projective planes; there are no additional relations between these free factors.

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