Cellular Homology, Singular Homology = Simplicial Homology

Simplicial Homology

Let K be a simplicial complex. Assume the vertices of K can be well ordered. (This is rarely a problem, and never a problem if you accept the axiom of choice.) Now the vertices of every simplex can be listed in order, and this ordering is consistent with intersection. If triangles abx and ayb share a common edge ab, a comes before b in each triangle, and in the common edge. This allows us to turn each simplex into a singular simplex. Map the vertices of δ_n onto the vertices of an n simplex, in order, and map everything else linearly, using barycentric coordinates.

We can generalize the above by mapping the vertices of δ_n to some, but not all, of the vertices of a simplex in K. Naturally some of the vertices will repeat, but they must remain in order. Thus all of δ₃ could map onto x.

Take the singular boundary of the aforementioned triangles abx and ayb. In each case the boundary includes the singular simplex ab, with a coefficient of +1 or -1. In other words, the boundary of a simplex becomes an alternating sum of simplexes that are already on our list. If these simplexes act as generators for free groups, we can build a chain complex for K, and derive its simplicial homology. This is a subchain of the original singular chain complex. After all, the singular simplex abx is an example of a singular simplex in the space K.

Simplicial = Cellular = Singular

Although our subchain is drastically smaller, it still generates the singular homology. Since cellular homology equals singular homology, it is sufficient to prove simplicial homology equals cellular homology. Remember that K is a cw complex, so it is reasonable to talk about its cellular homology. At each dimension, let closed cells and simplexes correspond. The homology of the skeleton K_p, relative to K_p-1, is generated by the p dimensional closed cells, which are precisely the p simplexes. If δ_p maps to a lower dimension, it lives in the face of a p cell, which is the boundary, hence it is already 0 in the relative homology. Take the singular boundary, and find the faces of a p simplex, with appropriate coefficients. Each of these is ± a generator in the lower relative homology. Take the quotient group and find the cellular homology, which is also the singular homology. The formula for simplicial homology is exactly the same. Therefore simplicial homology = cellular homology = singular homology.

Since everything maps back to singular homology, which depends only on K, the order of the vertices is not significant, and simplicial homology is well defined.

Just as cellular homology does not depend on the subdivision of a space into cells, so simplicial homology does not depend on the triangulation of a space into simplexes. Homology depends only on the space. In fact, you may have to find a space homeomorphic to S, before you can triangulate it into simplexes. Curved surfaces must become flat, and so on.

Simplicial Maps

A functor carries the category of simplicial complexes and simplicial maps into simplicial homology and homology homomorphisms. Since order doesn't matter, well order the vertices in the range space by assigning each vertex the least ordinal associated with the vertices in its preimage. Order the remaining vertices, that are not in the image of S, any way you like. As a continuous function from S into T, the simplicial map carries the simplex-based subchain of singular simplexes of S into the subchain of singular simplexes of T. This induces a homology homomorphism as usual.

If you want to extend this functor to all possible continuous functions from S into T, not just simplicial maps, switch to singular homology and apply the homomorphism there. Singular and simplicial homology are isomorphic in a natural way, so this can be converted to a homomorphism on simplicial homology. Alternatively one can use the simplicial approximation theorem to define a simplicial map, and an induced simplicial homology. Since any approximating function is homotopic to the original, the induced homology agrees with the singular homology, and is well defined.