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On $d$-dimensional cycles and the vanishing of simplicial homology
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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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Merging $\lim^1 \mathbf{A} \ne 0$ with other nonvanishing constructions
It is consistent that b=d=ω_n and lim^k A ≠ 0 for all 1≤k≤n, and that b=d=ω_{ω+2} with lim^k A ≠ 0 for every k≥1, by new forcings for lim^1 A ≠ 0 compatible with prior nonvanishing methods.
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