Structure on the Top Homology and Related Algorithms

TL;DR

This paper investigates the structure of top homology in compact triangulable spaces, revealing a canonical embedding into a free abelian group, which leads to a new invariant and an efficient basis construction algorithm.

Contribution

It introduces a canonical embedding of top homology into a free abelian group, establishing a matroid structure and providing a polynomial-time basis construction algorithm.

Findings

Top homology has a canonical embedding into a free abelian group.

The top homology structure forms an orientable matroid.

A polynomial-time algorithm for basis construction is developed.

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).