Structure on the top homology and related algorithms

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 Z coefficients).