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ガロア第一論文と乗数イデアル他関連資料スレ6
- 157 :132人目の素数さん:2024/01/25(木) 10:37:02.23 ID:zxKJrX2I.net
- >>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
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