On $d$-dimensional cycles and the vanishing of simplicial homology
In this paper we introduce the notion of a $d$-dimensional cycle which is a homological generalization of the idea of a graph cycle to higher dimensions. We examine both the combinatorial and homological properties of this structure and use these results to describe the relationship between the comb...
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| Main Author: | |
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| Format: | Journal Article |
| Language: | English |
| Published: |
29-11-2012
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| Subjects: | |
| Online Access: | Get full text |
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| Summary: | In this paper we introduce the notion of a $d$-dimensional cycle which is a
homological generalization of the idea of a graph cycle to higher dimensions.
We examine both the combinatorial and homological properties of this structure
and use these results to describe the relationship between the combinatorial
structure of a simplicial complex and its simplicial homology. In particular,
we show that over any field of characteristic 2 the existence of non-zero
$d$-dimensional homology corresponds exactly to the presence of a
$d$-dimensional cycle in the simplicial complex. We also show that
$d$-dimensional cycles which are orientable give rise to non-zero simplicical
homology over any field. |
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| DOI: | 10.48550/arxiv.1211.7087 |