Canonical Sphere Bases for Simplicial and Cubical Complexes

Sphere-bases are constructed for the ${\mathbb{Z}}_2$ vector space formed by the $k$-dimensional subcomplexes, of $n$-simplex (or $n$-cube), for which every $(k-1)$-face is contained in a positive even number of $k$-cells; addition is symmetric difference of the corresponding sets of $k$-cells. The bases consist of the boundaries of an algorithmically-specified family of $k+1$-simplexes or $k+1$-cubes. Geometric properties of these bases are investigated.