The Boundary Operator

Algebraic Topology · Axiom Academy

The key homomorphism connecting chain groups: mapping simplices to their oriented boundaries The boundary of an oriented n-simplex [v 0 , v 1 , ..., v n ] is the alternating sum of its (n-1)-dimensional faces, obtained by omitting each vertex in turn. 2. The Alternating Sign Formula The i-th face is obtained by removing the i-th vertex, and carries sign (-1) i . This alternating pattern ensures the boundary of a boundary equals zero. The boundary operator extends linearly to all chains: ∂(Σn i σ i ) = Σn i ∂(σ i ). Let's compute the boundary of a specific 2-simplex step by step, showing how each face contributes with its alternating sign. Notice that the three edges form a cycle: each vertex appears exactly twice with opposite orientations. 4. The Fundamental Property: ∂² = 0 The boundary of a boundary is always zero. Each (n-2)-face appears exactly twice in ∂(∂σ) with opposite signs, so they cancel out.

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