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arXiv · 2103.10830

Tri-Partitions and Bases of an Ordered Complex

Herbert Edelsbrunner·Katharina Ölsböck
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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.

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Herbert Edelsbrunner, Katharina Ölsböck. 2021-03-19. Tri-Partitions and Bases of an Ordered Complex. https://doi.org/10.1007/s00454-020-00188-x

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