連通和

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在數學裡，尤其是在拓撲學裡，連通和的運算是指一於流形上的幾何改變。其效果為將兩個給定的流形於各個選定的點附近連接起來。此一建構在閉曲面分類上有著關鍵性的角色。

更一般地，也可以將流形和其子流形連接起來；此一廣義化通常稱為纖維和。另外還有在結上之連通和的一相關概念，其稱為結和或結的複合。

[编辑] 於一點上的連通和

兩個m維流形的連通和為一流形，其將兩個流形各刪去一個球，再將球面邊界黏在一起。

若兩個流形是可定位的，there is a unique connected sum defined by having the gluing map reverse orientation. 即使這建構使用到的球的選擇，但最後結果都會於同胚下統一。亦可以將此運算作用於光滑範疇上，而其結果也會於微分同構下統一。

連通和的運算標記為 $\#$ ；例如， $A \# B$ 即表示為A和B的連通和。

連通和的運算中有一球面 $S m$ 為單位元；亦即， $M \# S^m$ 會同胚(或微分同構)於M。

閉球面的分類，在拓撲學上的一基本及重大結果，其描述任一閉球面均可表示成一g個環面和k個實射影平面的連通和。

[编辑] Connected sum along a submanifold

設 $M 1$ 和 $M 2$ 為兩個光滑、可定向且相同維度的流形，及V為一光滑、封閉且可定向的流形，可內嵌成 $M 1$ 和 $M 2$ 的子流形。此外，再假設其存在一法叢的同構

$\psi: N_{M_1} V \to N_{M_2} V$

其將每一纖維的定向顛倒。然後，ψ便可導出一定向保留的微分同構

$N_1 \setminus V \cong N_{M_1} V \setminus V \to N_{M_2} V \setminus V \to N_{M_2} V \setminus V \cong N_2 \setminus V,$

其中，每一法叢 $N_{M_i} V$ 都會微分同構地和於 $M i$ 內V的鄰域 $N i$ 一致，且映射

$N_{M_2} V \setminus V \to N_{M_2} V \setminus V$

is the orientation-reversing diffeomorphic involution

$v \mapsto v / |v|^2$

on normal vectors. The connected sum of $M 1$ and $M 2$ along $V$ is then the space

$(M_1 \setminus V) \bigcup_{N_1 \setminus V = N_2 \setminus V} (M_2 \setminus V)$

obtained by gluing the deleted neighborhoods together by the orientation-preserving diffeomorphism. The sum is often denoted

$(M_1, V) \# (M_2, V).$

Its diffeomorphism type depends on the choice of the two embeddings of $V$ and on the choice of $ψ$ .

Loosely speaking, each normal fiber of the submanifold $V$ contains a single point of $V$ , and the connected sum along $V$ is simply the connected sum described the preceding section, performed along each fiber. For this reason, the connected sum along $V$ is often called the fiber sum.

The special case of $V$ a point recovers the connected sum of the preceding section.

[编辑] Connected sum along a codimension-two submanifold

Another important special case occurs when the dimension of $V$ is two less than that of the $M i$ . Then the isomorphism $ψ$ of normal bundles exists whenever their Euler classes are opposite:

$e(N_{M_1} V) = -e(N_{M_2} V).$

Furthermore, in this case the structure group of the normal bundles is the circle group $S O (2)$ ; it follows that the choice of embeddings can be canonically identified with the group of homotopy classes of maps from $V$ to the circle, which in turn equals the first integral cohomology group $H 1 (V)$ . So the diffeomorphism type of the sum depends on the choice of $ψ$ and a choice of element from $H 1 (V)$ .

A connected sum along a codimension-two $V$ can also be carried out in the category of symplectic manifolds; this elaboration is called the symplectic sum.

[编辑] Local operation

The connected sum is a local operation on manifolds, meaning that it alters the summands only in a neighborhood of $V$ . This implies, for example, that the sum can be carried out on a single manifold $M$ containing two disjoint copies of $V$ , with the effect of gluing $M$ to itself. For example, the connected sum of a two-sphere at two distinct points of the sphere produces the two-torus.

[编辑] Connected sum of knots

There is a closely related notion of the connected sum of two knots. In fact, if one regards a knot merely as a one-manifold, then the connected sum of two knots is just their connected sum as a one-manifold. However, the essential property of a knot is not its manifold structure (all knots are circles) but rather its embedding into the ambient space. So the connected sum of knots has a more elaborate definition that produces a well-defined embedding, as follows.

Consider a planar projection of each knot and suppose these projections are disjoint.

Find a rectangle in the plane where one pair of sides are arcs along each knot but is otherwise disjoint from the knots.

Now join the two knots together by deleting these arcs from the knots and adding the arcs that form the other pair of sides of the rectangle.

This procedure results in the projection of a new knot, the connected sum (or knot sum, or composition) of the original knots.

Under this operation, knots in 3-space form a commutative monoid with prime factorization, which allows us to define what is meant by a prime knot. Proof of commutativity can be seen by letting one summand shrink until it is very small and then pulling it along the other knot. The unknot is the unit. The trefoil knot is the simplest prime knot. Higher dimensional knots can be added by splicing the $n$ -spheres.

In three dimensions, the unknot cannot be written as the sum of two non-trivial knots. This fact follows from additivity of knot genus; another proof relies on an infinite construction sometimes called the Mazur swindle. In higher dimensions, it is possible to get an unknot by adding two nontrivial knots.

[编辑] 另見

[编辑] 參考文獻

Robert Gompf: A new construction of symplectic manifolds, Annals of Mathematics 142 (1995), 527-595
William S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, 1991. ISBN 0-387-97430-X.

来自“http://www.wikilib.com/wiki/%E9%80%A3%E9%80%9A%E5%92%8C”

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