---
tag: R1RQ3226WV
kind: definition
name: Simplicial homology
msc: [55N10, 55U10]
defines: C6J40QF82R
uses: [V8GZ0W1ZZ5, 0WJX8MGG37, 0Z424HK8CQ]
sources:
  - url: https://en.wikipedia.org/wiki/Simplicial_homology
---

## Definition

Let $K$ be a [[simplicial complex|V8GZ0W1ZZ5]] with a total order on its vertex set.
For $n \ge 0$ let $C_n(K)$ be the [[free abelian group|0WJX8MGG37]] with basis the $n$-simplices of $K$, and let $C_n(K) = 0$ for $n < 0$.
An $n$-simplex with vertices $v_0 < v_1 < \cdots < v_n$ is written $[v_0, \dots, v_n]$.
The *boundary map* $\partial_n \colon C_n(K) \to C_{n-1}(K)$ is the homomorphism given on basis elements by
$$\partial_n [v_0, \dots, v_n] = \sum_{i=0}^{n} (-1)^i [v_0, \dots, \widehat{v_i}, \dots, v_n],$$
where $\widehat{v_i}$ means that $v_i$ is omitted, and $\partial_0 = 0$.
Then $\partial_{n} \circ \partial_{n+1} = 0$, so $(C_\bullet(K), \partial)$ is a chain complex.
The *$n$-th simplicial homology group* of $K$ is
$$H_n(K) = \ker \partial_n / \operatorname{im} \partial_{n+1}.$$

## Remarks

Elements of $\ker \partial_n$ are $n$-cycles and elements of $\operatorname{im} \partial_{n+1}$ are $n$-boundaries; the [[kernel|0Z424HK8CQ]] contains the image because $\partial_n \circ \partial_{n+1} = 0$.
The groups do not depend on the vertex order up to isomorphism.
With coefficients in an abelian group $A$, one uses $C_n(K) \otimes A$.

Example: let $K$ be the boundary of a triangle, with vertices $0 < 1 < 2$ and edges $[0,1]$, $[0,2]$, $[1,2]$.
Then $\partial_1 [0,1] = [1] - [0]$, $\partial_1 [0,2] = [2] - [0]$, $\partial_1 [1,2] = [2] - [1]$.
The kernel of $\partial_1$ is generated by $[0,1] - [0,2] + [1,2]$, and there are no $2$-simplices, so $H_1(K) \cong \Z$; also $H_0(K) \cong \Z$.

Non-example: dropping the signs $(-1)^i$ does not give a chain complex over $\Z$.
For the $2$-simplex, the unsigned boundary of $[0,1,2]$ is $[1,2] + [0,2] + [0,1]$, whose unsigned boundary is $2([0] + [1] + [2]) \ne 0$.