The Euler characteristic, -matroids, and a Möbius function
arXiv:2210.12483
Abstract
We first give two new proofs of an old result that the reduced Euler characteristic of a matroid complex is equal to the Möbius 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.
28 pages