The homological discrete vector field (HDVF for short) is a generalization of the discrete gradient vector field—a basic concept in discrete Morse theory. It emphasises the combinatorial nature of the homology computation. You can read the original journal paper or the third chapter of my PhD dissertation for a detailed presentation.

A HDVF of a cubical complex is a pair of disjoint subsets of its cells (P,S) such that the restriction of the boundary operator ∂ from S to P is a bijection. The cells in P are called primary and the cells in S, secondary.

Hence, a trivial HDVF is (P, S) = (∅, ∅). A less trivial HDVF is (P, S) = ({σ}, {τ}), where σ is a face of τ.

A HDVF defines a reduction between its chain complex and the chain complex generated by the cells that are not in the HDVF (called critical cells). This means that a maximal HDVF encodes the homology (and cohomology) groups of the complex:

The number of critical cells are the number of holes

The map g gives a homology basis

The map f* gives a cohomology basis

We define five elementary operations to transform a HDVF:

A takes a pair of critical cells and puts them into the HDVF (one in P, the other in S)

R takes a primary and a secondary cell of the HDVF and makes them critical

M swaps a primary cell with a critical cell

W swaps a secondary cell with a critical cell

MW swaps a primary cell with a secondary cell

Interactive tool

This demo allows you to play with a HDVF. Make a cubical complex and build an HDVF on it.

There is also a 3D version, but it is more complex to use.