The boundary of $\Delta^n$ is

The interior of $\Delta^n$ is $\Delta^n \setminus \partial \Delta^n$.

A $\Delta$-complex is a topological space $X$ with a family of maps

for each $n \ge 0$ such that ($\Delta^n$ is the standard $n$-simplex)

1. Each interior restriction of map $\sigma^n_\alpha$ is injective
2. Each point $x \in X$ lies in the image of interior restriction of $\sigma^n_\alpha$ for exactly one $\sigma^n_\alpha$
3. Each $i$-boundary restriction of $\sigma^n_\alpha$ is equal to some $\sigma^{n-1}_\beta$ where we identify the $i$-boundary with a $\Delta^{n-1}$ by the unique linear homeomorphism that preserves the order of the vertices
4. A subspace $A \subseteq X$ is open if and only if $(\sigma^n_\alpha)^{-1}(A) \subseteq \Delta^n$ is open for all $\sigma^n_\alpha$

A $\Delta$-complex $X$ comes with a filtration by skeleta $X^0 \subseteq \cdots \subseteq X^k \subseteq \cdots \subseteq X$ where the $k$-skeleton $X^k$ is the union of all images of simplices of dimension at most $k$.

A $\Delta$-complex $X$ is called $k$-dimensional if $X^k = X$ and $X^{k-1} \ne X$; and it is called infinite dimensional if there is no such $k$.

We say that $X$ is of finite type if for each $k \ge 0$, it has only finitely many $k$-simplices. We say $X$ is finite if it has only finitely many simplices altogether.

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

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

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

The boundary homomorphism $\partial_n : C^\Delta_n(X) \to C^\Delta_{n-1}(X)$Definition$n$-th simplicial chain module and $n$-chains2026-10-04 · Lîm Tsú-thuàn for $n \ge 1$ is defined on the basis by

i.e. Alternatively add each restriction $\sigma^n_\alpha\vert_{[v_0, \dots, \widehat{v}_i, \dots, v_n]}$. Recall the third condition of $\Delta$-complexDefinitionDelta complex2026-10-04 · Lîm Tsú-thuàn.

The composition $C^\Delta_n(X) \xrightarrow{\partial_n} C^\Delta_{n-1}(X) \xrightarrow{\partial_{n-1}} C^\Delta_{n-2}(X)$DefinitionBoundary homomorphism (also known as differential)2026-10-04 · Lîm Tsú-thuàn is the zero map.

The $n$-th homology group of a chain complexDefinitionChain complex2026-10-04 · Lîm Tsú-thuàn $(C_\bullet, \partial_\bullet)$ is the $R$-module

The $n$-th simplicial homology group of a $\Delta$-complex $X$ is the $\mathbb{Z}$-module