有序建筑群的三党和基地 (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 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.

翻译：将连接的平面图分解为一棵树和一棵双树,我们证明这是典型的Helmholz-Hodge平滑矢量场分解的组合类比。具体地说,我们显示,对于每个多面建筑群,$K美元和每个维度,$p美元,每维,美元,每维,每维,每维,每维,每维,每维,每维,每维,每维,每维,每维,每维,每维,每维,每平方美元,每平方美元,每平方美元,每平方美元,每平方美元,每平方美元,每立一双树,每立方美元,每立一小细胞,每组,每组就各1美元,每组,每立方美元,每立方美元,每立一小块,每组就可分成1美元,每平方美元,每平方美元,每立一股,每组各取1美元,每平方美元,每组每组每组的每组每组每组每组一美元,每组每组一美元,每组每组一美元,每组每组一美元,每组每组一美元,每组一美元,每组每组每组一美元,每组一美元,每组一美元,每组一美元,每组为每组每组1美元,每组每组每组每组1美元,每组每组每组每组每组为每组每组每组,每组,每组,每组每组每组每组每组每组每组每组每组每组,每组,每组,每组每组每组1美元,每组1美元,每组1美元,每组为1美元,每组,每组,每组,每组,每组,每组,每组1美元,每组每组1美元,每组为每组为每组1美元,每组每组为每组1美元,每组为每组1美元,每组1美元,每组1美元,每组1美元,每组1美元,每组1美元,每组为1美元,每组1美元,每组1美元,每组1美元,每组