拓扑正则邻域
Topological regular neighborhoods
摘要：
这篇文章是三个高度有影响力的文章之一的拓扑流形写的罗伯特D。爱德华兹在1970年代，但从未出版。几何拓扑讲习班（http：//www.uwm.edu/craigg/workshopgtt.htm）的组织者在国家科学基金会的支助下，为编写这些文章的电子版提供了便利，以便向公众提供。2006年完成了第一个“同源球悬浮液”的制备。关于这个系列的更完整的介绍可以在那篇文章中找到（arXiv：math/0610573）。本文提出了一种拓扑正则邻域理论，它代表了PL中分段线性正则邻域（或块丛）在TOP中的完全模拟。最简单地说，局部平坦嵌入流形Q（BdM和BdQ为空）的流形M的拓扑正则邻域是闭流形邻域V，它将BdV和M同胚固定到某个真满射BdV-->M的映射柱上。主定理断言这样的邻域的存在性和唯一性，对于dim Q>7。一个应用是胞胞腔复形形的胞状满射是一个简单的同伦等价（首先由查普曼证明同胚）。对于这种邻域有横截性的概念，该理论也适用于拓扑流形中的局部弱嵌入多面体。
英文摘要：
This article is one of three highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's but never published. Organizers of the Workshops in Geometric Topology (http://www.uwm.edu/~craigg/workshopgtt.htm) with the support of the National Science Foundation have facilitated the preparation of electronic versions of these articles to make them publicly available. Preparation of the first "Suspensions of homology spheres" was completed in 2006. A more complete introduction to the series can be found in that article (arXiv:math/0610573). In the current paper, a theory of topological regular neighborhoods is described, which represents the full analogue in TOP of piecewise linear regular neighborhoods (or block bundles) in PL. In simplest terms, a topological regular neighborhood of a manifold M locally flatly embedded in a manifold Q (BdM and BdQ empty) is a closed manifold neighborhood V which is homeomorphic fixing BdV and M to the mapping cylinder of some proper surjection BdV-->M. The principal theorem asserts the existence and uniqueness of such neighborhoods, for dim Q>7. One application is that a cell-like surjection of cell complexes is a simple homotopy equivalence (first proved for homeomorphisms by Chapman). There is a notion of transversality for such neighborhoods, and the theory also holds for locally tamely embedded polyhedral in topological manifolds.
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