范坎彭阻塞及其亲属
The van Kampen obstruction and its relatives
摘要：
我们回顾了经典van Kampen障碍θ对n-多面体到R^{2n}中的可嵌入性的无上链处理，并考虑了θ的几个类似物和推广，包括在流形情况下由J. P·达克斯获得了以下结果。- θ的模2约化是不完全的，这回答了Sarkaria的一个问题。- θ的奇维类似物完全阻碍了给定的n-多面体在R^{2n+1}中的无链接嵌入性（=“内在无链接”）。- θ的“放大”单参数形式是奇异纽结的泛1型不变量，即R^3中具有有限个刚性横向二重点的纽结。我们用它来决定在简单的同调条款时，一个给定的整数值1型不变量的奇异节点承认一个积分箭头图（= Polyak-维罗）公式。- 在亚稳域上解决了Yashchenko的一个问题，得到了每个PL流形N，如果它不可嵌入到给定的R^m，m \ge 3（n+1）/2中，则它包含一个子集X，使得没有映射N \to R^m将X和N-X发送到不交集. - 我们详细阐述了McCrory对塞曼谱序列的分析，以几何地表征“k-余连通和局部k-余连通”多面体，我们将其嵌入R^{2n-k}中，其中k<（n-3）/2，扩展了Penrose-Whitehead-Zeeman定理。
英文摘要：
We review a cochain-free treatment of the classical van Kampen obstruction θto embeddability of an n-polyhedron into R^{2n} and consider several analogues and generalizations of θ, including an extraordinary lift of θwhich in the manifold case has been studied by J.-P. Dax. The following results are obtained. - The mod 2 reduction of θis incomplete, which answers a question of Sarkaria. - An odd-dimensional analogue of θis a complete obstruction to linkless embeddability (="intrinsic unlinking") of the given n-polyhedron in R^{2n+1}. - A "blown up" 1-parameter version of θis a universal type 1 invariant of singular knots, i.e. knots in R^3 with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integer-valued type 1 invariant of singular knots admits an integral arrow diagram (=Polyak-Viro) formula. - Settling a problem of Yashchenko in the metastable range, we obtain that every PL manifold N, non-embeddable in a given R^m, m\ge 3(n+1)/2, contains a subset X such that no map N\to R^m sends X and N-X to disjoint sets. - We elaborate on McCrory's analysis of the Zeeman spectral sequence to geometrically characterize "k-co-connected and locally k-co-connected" polyhedra, which we embed in R^{2n-k} for k<(n-3)/2 extending the Penrose-Whitehead-Zeeman theorem.
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