Simplicial homology

Summary

Simplicial homology assigns to a simplicial complex 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 of the underlying space, so it is a topological invariant even though its definition uses a triangulation.