Obstructions to the smoothing of piecewise-differentiable homeomorphisms
Obstructions to the smoothing of piecewise-differentiable homeomorphisms
复制标题
分段可微同胚平滑的障碍
DOI：
10.1090/s0002-9904-1959-10345-1
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发表时间：
1959
影响因子：
1.3
通讯作者：
J. Munkres
中科院分区：
文献类型：
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作者：
J. Munkres
自1956年约翰·米尔诺（John Milnor）的基础论文发表以来[1]，他在S7上构造了与标准S7非微分同构的可微结构，米尔诺、r·索恩（R. Thorn）等人又获得了几个关于可微结构的进一步结果。本文在一个障碍理论的框架内对这些结果进行了统一和推广。两个可微流形ikf， N（连通的，不一定紧致的）如果具有同构的C2三角剖分，则称为组合等价。如果M和N是微分同态的，则它们在组合上是等价的；我们寻求部分逆。设f表示w流形M和N的C2个三角形之间的线性同构；我们尝试在M的开单形的邻域中重新定义f，从n - 1维开始向下，使f可微。一步之后，f不再是C2三角剖分之间的线性同构；因此，我们必须为这个逐步过程的归纳假设制定更一般的条件。当满足以下条件时，同纯形f: M - >N称为微分同构模L，其中L是M的C2三角剖分的w-框架：(1)f是L的每个闭单纯形上的C2微分同构；(2)f *在L的切向量上是一一对应的；(3)M的细分足够精细，对于L的每个单纯形a， a和/(cr)都存在平面的坐标邻域。(4) f在M-L上属于C1类，Df有界且\Df\有界远离0，在M-L的任意子集上具有紧闭包。（Df是雅可比矩阵。）(5)设或为l的单纯形，取其为平面的坐标邻域<e:1>；设z表示平面上的坐标
Since the publication in 1956 of John Milnor's fundamental paper [l ] in which he constructs differentiable structures on S7 nondiffeomorphic to the standard one, several further results concerning differ entiable structures have been obtained by Milnor, R. Thorn, and others. This paper unifies and extends some of these results within the framework of an obstruction theory. Two differentiable manifolds ikf, N (connected, not necessarily compact) will be said to be combinatorially equivalent if they possess isomorphic C2 triangulations. If M and N are diffeomorphic, then they are combinatorially equivalent [5 ] ; we seek a partial converse. Let ƒ denote a linear isomorphism between C2 triangulations of the w-manifolds M and N; we attempt to redefine ƒ in neighborhoods of the open simplices of M, beginning with dimension n — 1 and working down, so as to make ƒ differentiable . After one step, ƒ is no longer a linear isomorphism between C2 triangulations ; hence one must formu late more general conditions for the induction hypothesis of this stepby-step procedure. The homeomorphism ƒ : M—>N is called a diffeomorphism mod L, where L is the w-skeleton of a C2 triangulation of M, if the following conditions are satisfied : (1) ƒ is a C2 diffeomorphism on each closed simplex of L. (2) ƒ* is one-to-one on the tangent vectors to L. (3) The subdivision of M is fine enough that for each simplex a of L, there are coordinate neighborhoods of a and/(cr) in which they are flat. (4) ƒ is of class C1 on M—L, with Df bounded and \Df\ bounded away from zero on any subset of M—L having compact closure. (Df is the Jacobian matrix.) (5) Let or be a simplex of L. Choose a coordinate neighborhood of â in which it is flat ; let z denote coordinates in the plane of