Local knots of $2$-spheres in $4$-manifolds
Local knots of $2$-spheres in $4$-manifolds
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$4$ 流形中 $2$ 球体的局部结
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发表时间：
1969
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通讯作者：
Shin’ichi Suzuki
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文献类型：
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作者：
Shin’ichi Suzuki
在本文中，我们只从组合的观点来讨论。我们用（fMM）表示一对流形，使得M是一个三角化的定向四维流形，fM是一个适当嵌入的定向二维流形作为M中的子复形，f是M在M4中的逐段线性嵌入。我们测量嵌入f在M的内点x处的局部结类型 * 如下，[1]，[3]。设St（fx，M）和St（x，M）分别表示fx和x在M中的闭星星邻域. St（fx，M）的边界 ** S=3St（fx，M）是具有从M的取向继承的取向的3-球面，并且边界S1-St（x，M）是具有从M的取向继承的取向的1-球面。fS在S中的嵌入的定向纽结型（记为（x））称为f在x处的嵌入的局部纽结型。当（x）是平凡类型时，我们可以说局部纽结类型为0或fM在fx处局部平坦（无纽结）。一个2-流形fM称为局部平坦的，如果它在每个点上都是局部平坦的。当（x）是非平凡型时，我们可以说fM在fx上局部纽结，或者fx是fM的局部纽结点。当然，也可以在边界点xe 3 M处测量局部结类型。在这种情况下，cl（3St（x;M）JM）和cl（3St（x，M）JM）分别是3-胞腔和1-胞腔，并且局部结类型是（1，3）-胞腔对的类型。在本文中，我们将只考虑其边界点都是局部平坦（unknotted）的嵌入。由于局部纽结点必须是对（fMM）的任何三角剖分中的顶点，因此局部纽结点总是孤立的。如果M是紧的，则只能有有限个局部纽结点。R. H.福克斯和J. W.米尔诺观察到“在什么条件下给定的结类型集合，..，n是2-球面S在4-空间R中的某种嵌入的局部纽结类型的集合？并定义了切片结类型，并表明一个集合，.，n个结类型可以作为2-球面的局部结类型的集合出现
Throughout this paper we will only be concerned from the combinatorial point of veiw. By (fMM) we denote a pair of manifolds such that M is a triangulated oriented 4-dimensional manifold and fM is a properly embedded oriented 2-dimensional manifold as a subcomplex in M and f is a piesewise linear embedding of M in M4. We measure the local knot type* of the embedding f at an interior point x of M as follows, [1], [3]. Let St(fx, M) and St(x, M) denote the closed star neighborhoods of fx in M and x in M respectively. The boundary** S=3St(fx, M) of St(fx, M0 is a 3-sphere with an orientation inherited from that of M, and the boundary Sl--St(x, M) is a 1-sphere with an orientation inherited from that of M. The oriented knot type (denote (x)) of the embedding of fS in S is called the local knot type of the embedding f at x. When (x) is of trivial type, we may say that the local knot type is 0 or that fM is locally fiat (unknotted) at fx. A 2-manifold fM is called locally fiat if it is locally flat at each of its points. When (x) is of non-trivial type, we may say that fM is locally knotted at fx or that fx is locally knotted point of fM. Of course the local knot type can also be measured at a boundary point x e 3M. In this case cl(3St(fx ;M) JM) and cl(3St(x,M) JM) are 3-cell and 1-cell respectively and the local knot type is a type of (1, 3)-cell pair. In this paper we shall consider only embeddings whose boundary points are all locally flat (unknotted). Since a locally knotted point must be a vertex in any triangulation of the pair (fMM) the locally knotted points are always isolated. If M is compact, there can be only a finite number of locally knotted points. R. H. Fox and J. W. Milnor observed "Under which condition can a given collection of knot types , ..., n be the set of local knot types of some embedding of a 2-sphere S in the 4-space R? and defined the slice knot types and showed that a collection , ..., n of knot types can occur as the collection of local knot types of a 2-sphere