Chain Groups

Algebraic Topology · Axiom Academy

Turning simplices into algebra: free abelian groups generated by n-dimensional simplices Given a simplicial complex K, the n-chain group C n (K) is the free abelian group generated by all n-simplices in K. Elements are formal sums with integer coefficients. A chain is a formal linear combination of simplices: c = n 1 σ 1 + n 2 σ 2 + ... + n k σ k , where each n i ∈ ℤ. Chains can be added and subtracted using coefficients. Think of a chain as assigning an integer "weight" or "multiplicity" to each simplex. Positive coefficients represent "forward" orientation, negative represent "backward." An oriented n-simplex is an n-simplex with an ordering of its vertices: [v 0 , v 1 , ..., v n ]. Different orderings give different orientations. Even permutations preserve orientation; odd permutations reverse it. Orientation is crucial for defining the boundary operator. We conventionally write -σ to denote σ with reversed orientation. The collection of all chain groups forms a sequence: ... → C 2 (K) → C 1 (K) → C 0 (K) → 0. This structure, connected by boundary operators (to be defined next), is called a chain complex .

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