Faster computations of discrete homology

Machine computation of the discrete homology of graphs has stopped at degree two.

We present an algorithm that reaches degree four.

It generates the singular cubes inductively, pairing cubes one degree down instead of filtering all set maps; quotients the chain modules by the hyperoctahedral group action, over a field of sufficiently large characteristic; and shrinks the graph beforehand using homotopy invariance.

The fourth homology group of the five-cycle, previously beyond the reach of machine computation, is computed in under two days.