Simplicial Homology Groups

Algebraic Topology · Axiom Academy

LESSON Simplicial Homology Groups The fundamental algebraic invariants: measuring holes by computing ker ∂ / im ∂ An n-cycle is a chain c ∈ C n (K) with ∂c = 0. An n-boundary is a chain c that equals ∂d for some d ∈ C n+1 (K). We denote the cycles by Z n = ker ∂ n and boundaries by B n = im ∂ n+1 . The n-th homology group H n (K) is defined as the quotient group of cycles modulo boundaries: H n (K) = Z n / B n = ker ∂ n / im ∂ n+1 . Elements of H n (K) are equivalence classes [c] of cycles, where two cycles c and c' are equivalent if c - c' is a boundary. These classes are called homology classes . Homology groups detect "holes" in different dimensions. H 0 counts connected components, H 1 counts 1-dimensional holes (loops), H 2 counts 2-dimensional voids (cavities), and so on. A cycle that is not a boundary represents a genuine hole—you cannot "fill it in" within the complex. 4. Example: Homology of a Circle For the circle S 1 triangulated as a 1-dimensional complex with n vertices and n edges forming a loop, we compute: H 0 (S 1 ) = ℤ (one component) and H 1 (S 1 ) = ℤ (one 1-dimensional hole).

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