Microbundles and smoothing
Microbundles and smoothing
复制标题
微束和平滑
DOI：
10.1016/0040-9383(65)90003-0
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发表时间：
1965
期刊：
影响因子：
--
通讯作者：
M. Rothenberg
中科院分区：
文献类型：
--
作者：
R. Lashof;M. Rothenberg
在本文中，我们关注以下问题。假设有一个包含在无界可微流形 M 中的无界组合流形 K。何时存在分段可微同构 h: M+ A4 使得 h (K) 是 M 的光滑子流形。显然 K 在 M 中具有向量丛邻域是必要的。我们的主要结果断言，如果 K 是一个 &rite 复形的同伦类型，则反之亦然。该定理可以被认为是一种精化定理。 凯恩斯-赫希 (Cairns-Hirsch) 提出的平滑理论基本定理的版本。虽然“经典”凯恩斯-赫希定理是我们的出发点，但概括起来并不简单。我们的基本工具是 Milnor [5] 提出的分段线性微束理论，并且我们假设熟悉这些注释。除了由 Mazur [4] 和 Milnor 的笔记得出的“经典”凯恩斯-赫希定理的强形式之外，我们还利用了已经出版的微分拓扑技术和结果。
IN THIS PAPER we concern ourselves with the following question. Suppose one has an unbounded combinatorial manifold Kcontained in an unbounded differentiable manifold M. When does there exist a piecewise differentiable isomorphism h: M+ A4 such that h (K) is a smooth submanifold of M. It is clearly necessary that K have a vector bundle neighborhood in M. Our main result asserts that if K is the homotopy type of a &rite complex the converse is true.This theorem can be thought of as a refined version of the fundamental theorem of smoothing theory due to Cairns-Hirsch. While the “classical” Cairns-Hirsch theorem is our takeoff point the generalization is not straightforward. Our basic tool is the theory of piecewise linear microbundles due to Milnor [5], and we assume familiarity with these notes. Aside from a strong form of the “classical” Cairns-Hirsch theorem due to Mazur [4] and the notes of Milnor, we utilize techniques and results in differential topology which have already appeared in print.