H-principle的bordism版本
The bordism version of the h-principle
摘要：
从Segal构造的观点来看，每一个具有相干操作的范畴都产生一个上同调理论。类似地，流形的光滑映射上的每个开放稳定微分关系$R$确定了上同调理论$k^*$和$h^*$ ;上同调理论$k^*$描述了$R$的解的不变量，而$h^*$描述了$R$的所谓稳定形式解的不变量。我们证明了h-原理的边界形式：对于相当任意的开稳定微分关系$R$，上同调理论$k^*$和$h^*$是等价的.进一步，我们确定了$h^*$的同伦类型.因此，我们表明，对于一个相当任意的开放稳定的微分关系$R$，机器的稳定同伦理论可以应用到执行明确的计算和确定不变量的解决方案。在微分关系的所有解都是映射的情况下，我们的构造相当于Pontrjagin-Thom构造。在覆盖微分关系的情况下，我们的结果等价于Barratt-Priddy-Quillen定理，该定理断言n元有限集的置换群$Σ_n$的分类空间$BΣ_n$的直接极限同调等价于无限闭路空间$Ω^{\infty}S^{\infty}$的每个路径分量.在维数为$d=2$的映射上施加浸没微分关系的情况下，上同调理论$k^*$和$h^*$是不等价的.尽管如此，我们的方法仍然适用，可以用来恢复马德森-韦斯定理（芒福德猜想）。
英文摘要：
In view of the Segal construction each category with a coherent operation gives rise to a cohomology theory. Similarly each open stable differential relation $R$ imposed on smooth maps of manifolds determines cohomology theories $k^*$ and $h^*$; the cohomology theory $k^*$ describes invariants of solutions of $R$, while $h^*$ describes invariants of so-called stable formal solutions of $R$. We prove the bordism version of the h-principle: The cohomology theories $k^*$ and $h^*$ are equivalent for a fairly arbitrary open stable differential relation $R$. Furthermore, we determine the homotopy type of $h^*$. Thus, we show that for a fairly arbitrary open stable differential relation $R$, the machinery of stable homotopy theory can be applied to perform explicit computations and determine invariants of solutions. In the case of the differential relation whose solutions are all maps, our construction amounts to the Pontrjagin-Thom construction. In the case of the covering differential relation our result is equivalent to the Barratt-Priddy-Quillen theorem asserting that the direct limit of classifying spaces $BΣ_n$ of permutation groups $Σ_n$ of finite sets of n elements is homology equivalent to each path component of the infinite loop space $Ω^{\infty}S^{\infty}$. In the case of the submersion differential relation imposed on maps of dimension $d=2$ the cohomology theories $k^*$ and $h^*$ are not equivalent. Nevertheless, our methods still apply and can be used to recover the Madsen-Weiss theorem (the Mumford Conjecture).
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