  In the space of C^k piecewise expanding unimodal maps, k>=1, we characterize
the C^1 smooth families of maps where the topological dynamics does not change
(the "smooth deformations") as the families tangent to a continuous
distribution of codimension-one subspaces (the "horizontal" directions) in that
space. Furthermore such codimension-one subspaces are defined as the kernels of
an explicit class of linear functionals. As a consequence we show the existence
of C^{k-1+Lip} deformations tangent to every given C^k horizontal direction,
for k>=2.
