Group Presentations

Algebraic Topology · Axiom Academy

Understanding the notation ⟨generators | relations⟩ and reading presentations from CW complexes A group presentation has the form: The group is the quotient of the free group on the generators by the normal closure of the relations. 2. Reading Presentations from CW Complexes For a connected CW complex X with one 0-cell: Z_n = ⟨a | aⁿ⟩ (one generator, one relation) F₂ = ⟨a, b | ∅⟩ (two generators, no relations) Fₙ = ⟨a₁, ..., aₙ | ∅⟩ (n generators, no relations) 3. Fundamental groups of surfaces: Torus: π₁(T²) = ⟨a, b | aba⁻¹b⁻¹⟩ ≅ Z × Z Klein bottle: π₁(K) = ⟨a, b | aba⁻¹b⟩ Real projective plane: π₁(RP²) = ⟨a | a²⟩ ≅ Z₂

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