simplicial homology

If a space is built by gluing triangles, tetrahedra, and their higher analogues edge-to-edge — a simplicial complex — there is a finite, hands-on way to compute its holes without ever touching the uncountable machinery of singular theory. Simplicial homology is that combinatorial recipe: you list the simplices, write down their boundaries with signs, and do linear algebra over the integers.

Concretely, fix an ordering of the vertices. An oriented n-simplex [v_0, ..., v_n] contributes a generator to the chain group C_n, which is now a free abelian group on the finitely many n-simplices of the complex. The boundary operator is the explicit alternating face sum d[v_0, ..., v_n] = sum over i of (-1)^i [v_0, ..., v_i-hat, ..., v_n], where the hat means that vertex is deleted. One checks d d = 0 by a direct sign bookkeeping, and then H_n = ker(d_n) / im(d_{n+1}) exactly as before — but now every group is finitely generated, so you can literally compute the ranks and torsion by reducing integer matrices to Smith normal form.

The deep fact, proved by Eilenberg and others, is that for a triangulable space the simplicial homology of any triangulation agrees with the singular homology of the underlying space, and is therefore independent of the triangulation chosen. This is what justifies all hand computations: you pick the most economical triangulation, grind out the matrices, and the answer is a genuine topological invariant. It is also the historical and conceptual root of the whole subject — homology was simplicial first, and singular theory came later to remove the triangulation hypothesis.

The honest limitation is the triangulation hypothesis itself. Not every space is a simplicial complex, and even for manifolds the existence of a triangulation is subtle (it can fail in high dimensions). Simplicial homology is the right tool when you already have a combinatorial model; for arbitrary spaces you fall back on singular or cellular homology.