Deformation of homeomorphisms on stratified sets-猫眼课题宝

分层集上同胚的变形

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1972

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By L. C. Siebenmann

THE E E O R E M 0.有限单纯复形X到自身上的同胚的拓扑群H（X）是局部可收缩的。这个结果并不适用于ENR（欧几里德邻域收缩）。为了证明这一点，我们从S3 = R3 uo o通过将R3中相互不相交的任意非胞腔弧序列中的每一条压缩到一点而得到空间X：，41，a2，A3。. . .使得每个Ai，n~> 1，都是R3中单位球中由向量（4 n，0，0）平移的相同野弧A的副本。这个X是一个ENR;根据Andrews和Curtis [4]的一个结果，确实XXR同胚于S aXR = R4 0.显然，这个紧性允许自同胚h：X ~ X在单位元附近任意地置换fA 1，A2，A3，.的象。但是没有这样的h与恒等式是同位素的，因为这些是孤立点，在这些点上Y不是流形。(See非流形的处理大致上依赖于一种方法，用于变形R”x cX上的同胚，cY是X上的开锥，一旦给出了这样一种方法，则R“+1 xX-。然后通过对X的深度的归纳来进行证明。这里X被看作是一个分层集合，深度是X的非空层的维数的最大差。分层集合对于证明是至关重要的，因为它们的开子集本身就是分层集合，并且通常深度较小。因此，它只会澄清问题，从一开始就处理适当的分层集。我借此机会介绍类愉快的分层集，可能是拓扑类似物的多面体在分段线性领域或托姆的分层集在可微领域。这种证明方法几乎自动地提供了强的相对和尊重的变形定理（W 4.3，W 5.10），其中一个反例（W 2.3.1）建议：

T H E O R E M 0. The topological group H(X) of homeomorphisms of a finite simplicial complex X onto itself is locally contractible. This result does not extend to ENR's (euclidean neighborhood retracts). To see this let a space X be obtained from S 3 = R 3 u o o by crushing to a point each of a sequence of mutually disjoint wild non-cellular arcs in R 3 :,41, a 2 , A 3 . . . . such that each A,, n~> 1, is a copy of the same wild arc A in the unit ball in R 3 translated by the vector (4n, 0, 0). This X is an ENR; indeed X x R is homeomorphic to S a x R = R 4 0 by a result of Andrews and Curtis [4]. Clearly this compactum admits self-homeomorphisms h : X ~ X arbitrarily near the identity which nontrivially permute the images o fA 1, A2, A3, .... But no such h is isotopic to the identity because these are isolated points at which Y fails to be a manifold. (See also the fish skeleton of w The treatment of non-manifolds rests roughly speaking on a method for deforming homeomorphisms on R" x cX, cY being the open cone on X, once one is given such a method on R "+1 xX-. Then the proof proceeds by induction on the depth of X. Here Xis regarded as a stratified set, and depth is the greatest difference of dimensions of nonempty strata of X. Stratified sets are vital to the proof because their open subsets are themselves stratified sets, and often of a lesser depth. Thus it will only clarify matters to deal from the outset with suitable stratified sets. I take this opportunity to introduce classes of pleasant stratified sets that may come to be the topological analogues of polyhedra in the piecewise-linear realm or of Thom's stratified sets in the differentiable realm. This technique of proof almost automatically provides strong relative and respectful deformation theorems (w 4.3, w 5.10), which a counterexample (w 2.3.1) suggests are

Deformation of homeomorphisms on stratified sets

分层集上同胚的变形

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