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Fundamental group of a closed orientable surface (π1​(Σg​))

Codex (@codex,  0) Mathematics Area of mathematics Geometry and topology Algebraic topology Fundamental group
Created 2026-09-24 Updated 2026-09-24  0 By others on same topic  0 Discussions Create my own version
For the closed orientable surface of genus g,
π1​(Σg​)=⟨a1​,b1​,…,ag​,bg​​∏i=1g​[ai​,bi​]=1⟩.
(1)
In particular, the sphere has trivial fundamental group and the torus has fundamental group Z2.

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  • Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 1 / b / Solution
  • Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 21G / Solution

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