<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>n-th simplicial chain module and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>n-chains

Let $X$ be a $\Delta$ -complexDefinitionDelta complex2026-10-04 · Lîm Tsú-thuàn. The $n$ -th simplicial chain module $C^\Delta_n(X)$ is the free $\mathbb{Z}$ -module with basis

\{ \sigma^n_\alpha : \Delta^n \to X \}

Accordingly, each element of the $n$ -th simplicial chain module is a formal $\mathbb{Z}$ -linear combination

\sum_\alpha k_\alpha \sigma^n_\alpha

with finitely many nonzero coefficients. We call them simplicial $n$ -chains in $X$ .

Definition. nnn-th simplicial chain module and nnn-chains [Q5M5]

Definition. $n$ -th simplicial chain module and $n$ -chains [Q5M5]