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We first give two new proofs of an old result that the reduced Eulercharacteristic of a matroid complex is equal to the M\"obius number of thelattice of cycles of the matroid up to the sign. The purpose has been to find amodel to establish an analogous result for the case of $q$-matroids and we finda relation between the Euler characteristic of the simplicial chain complexassociated to a $q$-matroid complex and the lattice of $q$-cycles of the$q$-matroid. We use this formula to find the complete homology over$\mathbb{Z}$ of this shellable simplicial complex. We give a characterizationof nonzero Euler characteristic for such order complexes. Finally, based onthese results we remark why singular homology of a $q$-matroid equipped withorder topology may not be effective to describe the $q$-cycles unlike theclassical case of matroids.

The Euler characteristic, $q$-matroids, and a M\"obius function