查看“︁卡比演算”︁的源代码

{{about|幾何拓樸學上的分析工具|任天堂電玩角色卡比與其相關的遊戲系列|星之卡比系列}}
{{Rough translation|time=2021-09-01T20:40:02+00:00}}
在數學上，'''卡比演算'''是一個在[[几何拓扑学]]中用三維球面上有限多的形變步驟（卡比形變，{{lang-en|kirby moves}}）的集合使{{le|框连接|framed link}}產生形變的方法。它以[[羅比恩·卡比]]之名命名。[[羅比恩·卡比]]證明了若M與N皆為[[三維流形]] ，且它們分別是從L和J這兩個框連結上進行{{le|Dehn手术|Dehn surgery}}所得的，則它們是[[同胚]]的，[[當且僅當]]L和J藉由一連串的卡比形變產生關聯。根據{{le|Lickorish-Wallace定理|Lickorish-Wallace theorem}}，任意[[閉包|閉合]]且[[可定向]]的三維流形皆可由對三維球面裡的某些連結進行Dehn手術得到。<!--英語原文：In mathematics, the Kirby calculus in geometric topology is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. It is named for Robion Kirby. He proved that if M and N are 3-manifolds, resulting from Dehn surgery on framed links L and J respectively, then they are homeomorphic if and only if L and J are related by a sequence of Kirby moves. According to the Lickorish-Wallace theorem any closed orientable 3-manifold is obtained by such surgery on some link in the 3-sphere.--> 

一個擴張的圖像和形變集合被用以描述四維流形。一個在三維球體中的框連結暗示著二維[[手柄]]對四維球的依附（此流形的三維邊界是上述連結圖的三維流形描述）。一維手柄可由兩個三維球（一維手柄的依附區）或（更常見地）有著點的非紐結化圓表示。這個點表示著一個標準有界的二維圓盤的鄰域，也就是有著點的圓，會被從四維球的內部切除。切除這個二維手柄相當於加上一個一維手柄。三維和四維的手柄通常不會在圖中指示出來。
<!--英語原文：An extended set of diagrams and moves are used for describing 4-manifolds. A framed link in the 3-sphere encodes instructions for attaching 2-handles to the 4-ball. (The 3-dimensional boundary of this manifold is the 3-manifold interpretation of the link diagram mentioned above.) 1-handles are denoted by either (a) a pair of 3-balls (the attaching region of the 1-handle) or, more commonly, (b) unknotted circles with dots. The dot indicates that a neighborhood of a standard 2-disk with boundary the dotted circle is to be excised from the interior of the 4-ball. Excising this 2-handle is equivalent to adding a 1-handle. 3-handles and 4-handles are usually not indicated in the diagram.--> 

== 手柄解構 == 

* 一個閉合平滑的四維流形M通常會用{{le|手柄结构|handle decomposition}}描述。
* 一個零維手柄就是一個球，而其{{le|依附映射|attaching map}}是不交並的。
* 一個一維手柄是沿著兩個不相交的三維[[球 (數學)|球]]依附的。
* 一個二維手柄是沿著一個{{le|立体环面|solid torus}}依附的。由於這個固體環嵌於一個三維流形中，四維流形上的手柄解構和三維流形上的[[紐結理論]]之間有關係。
* 一對指數相差1的手柄，當其中心以一個足夠簡單的方法連結時，它們可以消除而不會改變下面的流形。同樣，這些對可以創造。
在一個平滑四維流形中，兩個不同平滑手柄體（handlebody）的解構跟依附映射的[[同痕]]有限序列，以及手柄對的創造和消除（creation/cancellation）有關聯。 

<!--英語原文：
*A closed, smooth 4-manifold M is usually described by a handle decomposition. 
*A 0-handle is just a ball, and the attaching map is disjoint union. 
*A 1-handle is attached along two disjoint 3-balls. 
*A 2-handle is attached along a solid torus; since this solid torus is embedded in a 3-manifold, there is a relation between handle decompositions on 4-manifolds, and knot theory in 3-manifolds. 
*A pair of handles with index differing by 1, whose cores link each other in a sufficiently simple way can be cancelled without changing the underlying manifold. Similarly, such a cancelling pair can be created. 

Two different smooth handlebody decompositions of a smooth 4-manifold are related by a finite sequence of isotopies of the attaching maps, and the creation/cancellation of handle pairs. --> 

==參見== 

* [[Exotic R4|Exotic <math>\R^4</math>]] 

==參考文獻==
* Rob Kirby, "A Calculus for Framed Links in S<sup>3</sup>".  Inventiones Mathematicae, vol. 45, 1978, pgs. 35-56. 

* Robert Gompf and Andras Stipsicz, ''4-Manifolds and Kirby Calculus'', (1999) (Volume 20 in ''Graduate Studies in Mathematics''), American Mathematical Society, Providence, RI ISBN 0-8218-0994-6 

[[Category:几何拓扑学]]
[[Category:微积分]]