The Euler characteristic, $q$-matroids, and a M\"obius function
Abstract
We first give two new proofs of an old result that the reduced Euler characteristic of a matroid complex is equal to the M\"obius number of the lattice of cycles of the matroid up to the sign. The purpose has been to find a model to establish an analogous result for the case of -matroids and we find a relation between the Euler characteristic of the simplicial chain complex associated to a -matroid complex and the lattice of -cycles of the -matroid. We use this formula to find the complete homology over of this shellable simplicial complex. We give a characterization of nonzero Euler characteristic for such order complexes. Finally, based on these results we remark why singular homology of a -matroid equipped with order topology may not be effective to describe the -cycles unlike the classical case of matroids.
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Cite
@article{arxiv.2210.12483,
title = {The Euler characteristic, $q$-matroids, and a M\"obius function},
author = {Trygve Johnsen and Rakhi Pratihar and Tovohery Hajatiana Randrianarisoa},
journal= {arXiv preprint arXiv:2210.12483},
year = {2022}
}
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28 pages