ガロア第一論文と乗数イデアル他関連資料スレ6

[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ6 (1002ﾚｽ)
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157: １３２人目の素数さん [] 2024/01/25(木) 10:37:02.23 ID:zxKJrX2I

>>149 訂正と補足

＜訂正＞
・それが、ミルナーのh-cobordismにつながり
　　↓
・それが、Smaleのh-cobordismにつながり

(参考)
https://en.wikipedia.org/wiki/H-cobordism
h-cobordism
In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps
M → W and N → W
are homotopy equivalences.

The h-cobordism theorem gives sufficient conditions for an h-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder M × [0, 1]. Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds.

The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture.

Background
Before Smale proved this theorem, mathematicians became stuck while trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were even harder. The h-cobordism theorem showed that (simply connected) manifolds of dimension at least 5 are much easier than those of dimension 3 or 4. The proof of the theorem depends on the "Whitney trick" of Hassler Whitney, which geometrically untangles homologically-tangled spheres of complementary dimension in a manifold of dimension >4. An informal reason why manifolds of dimension 3 or 4 are unusually hard is that the trick fails to work in lower dimensions, which have no room for entanglement.
(引用終り)

＜補足＞
　>>149の福田拓生先生が、ルネ・トムから直接聞いた話の素直な解釈は
　H.Cartan：岡論文を読め
　　↓
　岡論文：上空移行 次元を上げよ
　　↓
　ルネ・トム：岡先生ありがとう、+１次元のコボルディズムが閃いた
　　↓
　ルネ・トム：「日本で岡先生に会えたときには感激した」と懐かしそうに言われた

ということではないでしょうか

http://rio2016.5ch.net/test/read.cgi/math/1704672583/157

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