Quantifying the homology of periodic cell complexes
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866914154475945984 |
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| author | Onus, Adam Robins, Vanessa |
| author_facet | Onus, Adam Robins, Vanessa |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_09223 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quantifying the homology of periodic cell complexes Onus, Adam Robins, Vanessa Algebraic Topology Geometric Topology 55-08, 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. |
| title | Quantifying the homology of periodic cell complexes |
| topic | Algebraic Topology Geometric Topology 55-08, |
| url | https://arxiv.org/abs/2208.09223 |