# Speeding up Homology via Duality?

In the last post I constructed for any (locally finite) simplicial complex $K$ a dual complex. Today, I thought of using this to gain a potential speed-up in computational homology.

Note that simplicial homology of $K$ is the same as Cech homology of the cover given by open stars of the vertices of $K$. This is because the nerve of this cover is precisely $K$ and Cech homology is not-so-secretly the simplicial homology of the nerve.

However, we know that the dual complex has the same homotopy type as $K$, but the number of vertices in the dual complex is equal to the number of maximal simplices. Simplicial homology of this dual complex is the same as Cech homology of the cover given by the closed top dimensional cells in $K$.

Here is a case where this observation results in a most dramatic speed-up. The standard n-simplex $\Delta^n$ has $2^{n+1}$ simplices and naively simplicial homology requires storing all of these. However, the dual complex is a single point and Cech homology here reveals that. Thus we have gone from a chain complex with $n+1$ chain groups, each of which are dimension $\binom{n+1}{j}$ to a chain complex concentrated in a single degree, all by using this duality.

Of course, the case of graphs shows where this duality fails miserably. The dual complex creates simplices whose dimension corresponds to the degree of vertices in the original graph, so we can go from a two-term chain complex with a single boundary matrix, to arbitrarily large chain complexes.

This presents a question of possible interest: For what simplicial complexes does the dual complex and the original complex have the same number of simplices? Triangulated manifolds clearly do, but what about other simplicial complexes? Can we say something about the expected discrepancy of these numbers?