Cell-like mappings and their generalizations
Cell-like mappings and their generalizations
复制标题
类细胞映射及其概括
DOI：
10.1090/s0002-9904-1977-14321-8
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发表时间：
1977
影响因子：
1.3
通讯作者：
R. Lacher
中科院分区：
文献类型：
--
作者：
R. Lacher
胞状映射是那些点逆是胞状空间（作为域的子空间）的映射。一个空间是类胞腔的，如果它同胚于某个流形的胞腔子集。这个定义是在1968年给出的，当时我开始研究欧氏邻域收缩（ENR）之间的适当的胞状映射。当时，我指出，这样的地图形成一个类别，其中包括适当的，满射，多面体和适当的，从流形到ENR细胞地图之间的可收缩的地图。S. Smale以前研究过ANR之间的可压缩映射，证明了一个类似Vietoris的定理;科恩已经证明了有限多面体之间的PL可收缩映射是简单的同伦等价;当然，流形上的适当的细胞映射已经被广泛研究（“点状分解”）。这种统一在当时看来是值得的，因为胞状（对于ENR之间的真满射映射）和D。沙利文在与Hauptvermutung：限制任何开集的逆是一个适当的同伦等价。使用沙利文的工作，它遵循的细胞之间的PL流形映射往往是同伦的PL同胚。1971年，L. C. Siebenmann在上面的句子中发现了“PL”的可疑的红鲱鱼性质：胞状映射的集合M -> N（其中M和N是闭的n-流形，f * T M）正是映射M -+ N空间中同胚集合M -> N的闭包。(The情况n = 3也要求M不包含假立方体，并且S之前已经完成了。Armentrout.）“细胞样”概念从此被研究、推广和类比。我将尝试叙述一些最近的工作。三个主要议题是确定的：A.直接性定理及其全局结果; B.映射圆柱邻域;和C.空间的去奇异化。最近的重大发现可以被认为至少是外围的主题：简单同伦型的拓扑不变性（由T。Chapman），紧ANR的有限性（J. West），以及某些同调球面的双悬置的局部欧几里德性质（R.
Cell-like maps are those whose point-inverses are cell-like spaces (as subspaces of the domain). A space is cell-like if it is homeomorphic to a cellular subset of some manifold. This definition was given in 1968, at which time I began to study proper, cell-like maps between euclidean neighborhood retracts (ENR's). At the time, I pointed out that such maps form a category which includes proper, surjective, contractible maps between polyhedra and proper, cellular maps from a manifold to an ENR. S. Smale had previously studied contractible maps between ANR's, proving a Vietoris-like theorem; M. Cohen had shown that PL contractible maps between finite polyhedra are simple homotopy equivalences; and, of course, proper, cellular maps on manifolds had been studied extensively ("point-like decompositions"). This unification seemed worthwhile at the time because of the equivalence of cell-like (for proper, surjective maps between ENR's) and a condition studied by D. Sullivan in connection with the Hauptvermutung: the restriction to the inverse of any open set is a proper homotopy equivalence. Using Sullivan's work it followed that a cell-like map between PL manifolds is often homotopic to a PL homeomorphism. In 1971, L. C. Siebenmann identified the suspected red herring nature of "PL" in the above sentence: The set of cell-like maps M -> N (where M and N are closed «-manifolds, /* T M ) is precisely the closure of the set of homeomorphisms M -> N in the space of maps M -+ N. (The case n = 3 requires also that M contain no fake cubes and was done earlier by S. Armentrout.) The "cell-like" concept has since been studied, generalized, and analogized. I will attempt to recount some of this recent work. Three main topics are identifiable: A. Finiteness theorems and their global consequences; B. Mapping cylinder neighborhoods; and C. Desingularizations of spaces. Recent major discoveries can be considered at least peripheral to the theme: The topological invariance of simple homotopy type (by T. Chapman), the finiteness of compact ANR's (by J. West), and the locally euclidean nature of the double suspension of certain homology spheres (by R.