Tri-Partitions and Bases of an Ordered Complex

TL;DR

This paper extends graph decomposition concepts to polyhedral complexes, establishing a unique tri-partition of cells that facilitates the computation of canonical bases for homology groups.

Contribution

It introduces a tri-partition of cells in polyhedral complexes, generalizing graph decompositions and providing a matrix reduction algorithm for basis construction.

Findings

Tri-partition of p-cells into maximal p-tree, p-cotree, and Betti number cells.

Unique tri-partition for ordered p-cells.

Algorithm for basis construction of cycle and boundary groups.

Abstract

Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholz-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 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.