Failure of the parametric h-principle for maps with prescribed jacobian

Joseph Coffey

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arXiv · math/0512126

Failure of the parametric h-principle for maps with prescribed jacobian

Joseph Coffey

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Abstract

Let M and N be closed n-dimensional manifolds, and equip N with a volume form σ. Let μbe an exact n-form on M. Arnold then asked the question: When can one find a map f:;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_bdl:T M to T N which is degree 0 on the base and induces μone 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 μfrom σ, 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.

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Joseph Coffey. 2005-12-06. Failure of the parametric h-principle for maps with prescribed jacobian. https://arxiv.org/abs/math/0512126

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