具有指定Jacobian映射的参数h-原理的失效
Failure of the parametric h-principle for maps with prescribed jacobian
摘要：
设M和N是n维闭流形，且N具有体积形式σ.设μ是M上的正合n-形式。阿诺德然后问了一个问题：什么时候可以找到一个映射f：;N，使得f*σ=μ。在1973年Eliashberg和Gromov表明，这个问题是，在深层次意义上，平凡：它满足h-原则，每当一个人可以找到一个丛映射f_bdl：T M到T N这是度0的基础上，并导致μ一个可以homotop这个地图的解决方案f。也就是说，如果满足朴素拓扑条件，则可以找到解。在这个问题中没有进一步有趣的几何学。我们证明了相应的参数h-原理失效--如果考虑从σ导出μ的映射族，可以在解空间中找到h-原理所不能预测的有趣拓扑。此外，这种映射的同伦类型是量子化的：对于某些形式族，同伦类型保持不变，只在离散值处跳跃。
英文摘要：
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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