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arXiv · 1203.5288

Structure on the Top Homology and Related Algorithms

Nissim Ranade·Chandrika Sadanand·Dennis Sullivan
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Abstract

We explore the special structure of the top-dimensional homology of any compact triangulable space $X$ of dimension $d$. Since there are no $(d+1)$-dimensional cells, the top homology equals the top cycles and is thus a free abelian group. There is no obvious basis, but we show that there is a canonical embedding of the top homology into a canonical free abelian group which has a natural basis up to signs. This embedding structure is an invariant of $X$ up to homeomorphism. This circumstance gives the top homology the structure of an (orientable) matroid, where cycles in the sense of matroids correspond to the cycles in the sense of homology. This adds a novel topological invariant to the topological literature. We apply this matroid structure on the top homology to give a polynomial-time algorithm for the construction of a basis of the top homology (over $\mathbb{Z}$ coefficients).

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Nissim Ranade, Chandrika Sadanand, Dennis Sullivan. 2019-11-27. Structure on the Top Homology and Related Algorithms. https://arxiv.org/abs/1203.5288

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