Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 21F / a

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- Solution a

Solution

The Seifert-van Kampen theorem says that if a path-connected space is the union of path-connected open sets

U, V

with path-connected intersection, then

π_{1} (U \cup V) = π_{1} (U) * π_{1} (V) / ⟨ ⟨ i_{U *} (w) i_{V *} (w)^{- 1} : w \in π_{1} (U \cap V)⟩ ⟩ .

Using the usual polygonal cell decompositions, the torus attaches a two-cell to

S^{1} \lor S^{1}

along

ab a^{- 1} b^{- 1}

, while the projective plane attaches one along

a^{2}

. Hence

π_{1} (T^{2}) = ⟨ a, b ∣ ab a^{- 1} b^{- 1} ⟩ ≅ Z^{2}, π_{1} (RP^{2}) = ⟨ a ∣ a^{2} ⟩ ≅ Z /2 .

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