Failure of parametric H-principle for maps with prescribed Jacobian

Let M and N be closed n-dimensional manifolds, and equip N with a volume form σ. Let μ be an exact n-form on M. Arnold then asked the question: When can one find a map f:M→N such that f^∗σ=μ. 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_{b d l}:T M→T N which is degree 0 on the base and such that f_{b d l}^∗(σ)=μ 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 μ from σ, one can find interesting topology in the space G_μ 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.