Tri-partitions and Bases of an Ordered Complex

Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K , and every dimension, p , there is a partition of the set of p -cells into a maximal p -tree, a maximal p -cotree, and a collection of p -cells whose cardinality is the p -th reduced Betti number of K . Given an ordering of the p -cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.