---
tag: C6J40QF82R
kind: concept
name: Simplicial homology
msc: [55N10, 55U10]
sources:
  - url: https://en.wikipedia.org/wiki/Simplicial_homology
same_as:
  wikidata: Q7520902
  wikipedia: Simplicial homology
---

## Summary

Simplicial homology assigns to a [[simplicial complex|V8GZ0W1ZZ5]] a sequence of abelian groups $H_n(K)$ that count its $n$-dimensional holes.
For a finite complex it is computed by finite linear algebra over $\Z$.
Chains are formal sums of oriented simplices, and the boundary operator sends a simplex to the signed sum of its faces.
It agrees with the [[singular homology|M0Q5CFJ7ZH]] of the underlying space, so it is a topological invariant even though its definition uses a triangulation.

## Explanations

### L1 human

A cycle is a combination of simplices with no boundary, like a loop of edges.
A boundary is a cycle that is filled in by higher simplices, like the edge loop around a triangle that is present in the complex.
The $n$-th homology group counts cycles that are not boundaries.
For the hollow triangle, made of three edges, the loop around it is a cycle that is not filled, so $H_1 \cong \Z$.
If the triangle is filled in, the loop becomes a boundary and $H_1 = 0$.