  Let M and N be closed n-dimensional manifolds, and equip N with a volume form
\sigma. Let \mu be an exact n-form on M. Arnold then asked the question: When
can one find a map f:;N such that f*\sigma=\mu. In 1973 Eliashberg and Gromov
showed that this problem is, in a deep sense, trivial: It satisfies an
h-principle, and whenever one can find a bundle map f_bdl:T M to T N which is
degree 0 on the base and induces \mu one can homotop this map to a solution f.
That is if the naive topological conditions are satisfied on can find a
solution. There is no further interesting geometry in the problem.
  We show the corresponding parametric h-principle fails- if one considers
families of maps inducing \mu from \sigma, one can find interesting topology in
the space of solutions which is not predicted by an h-principle. Moreover the
homotopy type of such maps is quantized: for certain families of forms homotopy
type remains constant, jumping only at discrete values.
