  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.
