论文标题
量化周期性细胞复合物的同源性
Quantifying the homology of periodic cell complexes
论文作者
论文摘要
周期性的单元格综合体$ k $，具有有限的代表作为商空间，$ q（k）$，由在$ k $上作用的翻译组中标识的等效类单元组成。我们研究$ K $的贝蒂数字和周期与$ Q（k）$的赛车有关，首先是$ k $是图形，然后是较高尺寸的细胞复合物。当$ k $是$ d $ - 周期图时，可以在商图的边缘定义$ \ mathbb {z}^d $ - weights，并且此信息可以完全恢复$ k $的同源性生成器。高维细胞复合物的情况更加微妙，并使用Mayer-Vietoris光谱序列进行了详细的研究。
A periodic cell complex, $K$, has a finite representation as the quotient space, $q(K)$, consisting of equivalence classes of cells identified under the translation group acting on $K$. We study how the Betti numbers and cycles of $K$ are related to those of $q(K)$, first for the case that $K$ is a graph, and then higher-dimensional cell complexes. When $K$ is a $d$-periodic graph, it is possible to define $\mathbb{Z}^d$-weights on the edges of the quotient graph and this information permits full recovery of homology generators for $K$. The situation for higher-dimensional cell complexes is more subtle and studied in detail using the Mayer-Vietoris spectral sequence.
