Problems on homology manifolds
Problems on homology manifolds
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同调流形问题
DOI：
10.2140/gtm.2006.9.87
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
F. Quinn
中科院分区：
文献类型：
--
作者：
F. Quinn
57P99在这些注释中，“同调流形”是指ENR(欧几里得邻域收缩)Z系数同调流形，除非另有说明，而“奇异”是指不是流形因子(即局部或“Quinn”指标⁄1。我们使用局部指标的乘法版本，取值于1C8Z)。在过去的十年里，奇异的同调流形被证明是存在的，并发展了相当多的结构理论。然而，它们还没有出现在数学的其他领域。第一组问题提出了可能发生这种情况的方法。后面的问题更多地是这个主题的内在问题。第一节、第二节和第三节涉及同调流形可能的“自然”外观：作为非球面几何对象；作为Gromov-Hausdorff极限；作为紧致化的边界。第四节讨论群作用，其中使用同调流形不动集可以给出更简单的分类结果。第5节和第6节考虑对非ANR和“近似”同调流形的可能推广。第7节涉及具有特殊度量结构的空间。第8节描述了当前理论中仍未解决的低维案例。第9节收集了与同胚和奇异同调流形的“不相交圆盘性质”有关的问题。
57P99 In these notes “homology manifold” means ENR (Euclidean neighborhood retract) Z‐coefficient homology manifold, unless otherwise specified, and “exotic” means not a manifold factor (ie local or “Quinn” index⁄1. We use the multiplicative version of the local index, taking values in 1C8Z). In the last decade exotic homology manifolds have been shown to exist and quite a bit of structure theory has been developed. However they have not yet appeared in other areas of mathematics. The first groups of questions suggest ways this might happen. Later questions are more internal to the subject. Section 1, Section 2, and Section 3 concern possible “natural” appearances of homology manifolds: as aspherical geometric objects; as Gromov‐Hausdorff limits; and as boundaries of compactifications. Section 4 discusses group actions, where the use of homology manifold fixed sets may give simpler classification results. Section 5 and Section 6 consider possible generalizations to non-ANR and “approximate” homology manifolds. Section 7 concerns spaces with special metric structures. Section 8 describes still-open low dimensional cases of the current theory. Section 9 collects problems related to homeomorphisms and the “disjoint disk property” for exotic homology manifolds.