On $d$-dimensional cycles and the vanishing of simplicial homology

Emma Connon

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arXiv · 1211.7087

On $d$-dimensional cycles and the vanishing of simplicial homology

Emma Connon

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Abstract

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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BibTeX RIS

Emma Connon. 2013-07-22. On $d$-dimensional cycles and the vanishing of simplicial homology. https://arxiv.org/abs/1211.7087

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