>>762
>＞ところが、５次元多様体ではsurgery theory（手術理論）による分類が可能！
>　これまた753で書いた通り
>　手術理論はｈ同境定理の証明で用いられるが
>　その条件を見れば、単連結と書いてある
>　したがって、任意の多様体ではなく、単連結多様体の分類
>【定理 4.7 (Smale:可微分多様体のh 同境定理)】
>V, V を連結かつ『単連結』な閉じた n次元 C∞ 多様体とする．
>もしも，n ≥ 5 であって V とV が h 同境ならば，
>V と V は C∞ 同相である．

定理を読み違えているよ
１）まず 定理 4.7 (Smale:可微分多様体のh 同境定理) 『単連結』とあるが
　”可微分”を見落としているんじゃないの？
２）いま主に問題にしている４次元キャッソンハンドルの話は、”可微分”ではない
　実際、h-cobordism は、下記引用の通り『単連結』のしばりなし（５次元で参考にする例も同じ）

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.

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