非稳定Nielsen重合不变量与Hopf-Ganea同态
Nonstabilized Nielsen coincidence invariants and Hopf--Ganea homomorphisms
摘要：
在经典的不动点和重合点理论中，Nielsen数的概念已被证明是非常富有成果的。我们利用非稳定法边理论作为主要工具，将其推广到任意维流形之间的映射对（f_1，f_2）。这导致路径分量（和点）的最小数目MCC（f_1，f_2）（和MC（f_1，f_2））的估计。在与（f_1，f_2）同伦的映射对的重合集上。进一步，我们推导出MC（f_1，f_2）的有限性条件。作为一个应用程序，我们计算这两个最小的数字明确在各种具体的几何样本的情况下。重合集的Nielsen分解是通过将某一道路空间E（f_1，f_2）分解为道路分量而得到的。其更高维度的拓扑结构捕获了更重要的几何重合数据。在设置的同伦群所产生的不变量密切相关的某些霍普夫-Ganea同态，从而产生有限性障碍MC。
英文摘要：
In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs (f_1,f_2) of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbers MCC(f_1,f_2) (and MC(f_1,f_2), respectively) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to (f_1,f_2). Furthermore, we deduce finiteness conditions for MC(f_1,f_2). As an application we compute both minimum numbers explicitly in various concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E(f_1,f_2) into path components. Its higher dimensional topology captures further crucial geometric coincidence data. In the setting of homotopy groups the resulting invariants are closely related to certain Hopf--Ganea homomorphisms which turn out to yield finiteness obstructions for MC.
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