  We develop exact piecewise polynomial sequences on Alfeld splits in any
spatial dimension and any polynomial degree. An Alfeld split of a simplex is
obtained by connecting the vertices of an $n$-simplex with its barycenter. We
show that, on these triangulations, the kernel of the exterior derivative has
enhanced smoothness. Byproducts of this theory include characterizations of
discrete divergence-free subspaces for the Stokes problem, commutative
projections, and simple formulas for the dimensions of smooth polynomial
spaces.
