Attaching Cells

Algebraic Topology · Axiom Academy

CW complexes and how attaching 2-cells affects the fundamental group A CW complex is a topological space built by attaching cells of increasing dimension: The 1-skeleton X¹ consists of all 0-cells and 1-cells. This determines the fundamental group structure before 2-cells are attached. To attach a 2-cell e² to a space X, we specify an attaching map φ: S¹ → X that determines how the boundary of the disk glues to the existing space. The new space is formed as a pushout: X ∪_φ D² 3. Effect on the Fundamental Group Example: Start with S¹ ∨ S¹ (figure-eight), which has π₁ = F₂ = ⟨a, b⟩. If we attach a 2-cell along the loop aba⁻¹b⁻¹ (the commutator), we "fill in" this loop, making it contractible. Specific Example: Attaching a 2-cell along aba⁻¹b⁻¹ gives: π₁ = ⟨a, b | aba⁻¹b⁻¹⟩ ≅ Z × Z (this is the torus!)

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