Decomposing Euclidean space with a small number of smooth sets

Let the cardinal invariant ${\mathfrak s}_{n}$ denote the least number of continuously smooth $n$-dimensional surfaces into which $(n+1)$-dimensional Euclidean space can be decomposed. It will be shown to be consistent that ${\mathfrak s}_{n}$ is greater than ${\mathfrak s}_{n+1}$. These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.