2012/11/29 by Emma Connon, Connon, Emma
Computer Science · Mathematics · #Commutative Algebra and Its Applications #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.AC #math.AT #math.CO
paper · pdf · doi:10.48550/arxiv.1211.7087
Substantially revised and expanded
arxiv created 2013/07/22 · arxiv updated 2013/07/23
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.