A characterization of groups of closed orientable surfaces in 4-space
A characterization of groups of closed orientable surfaces in 4-space
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4 空间中封闭可定向曲面组的表征
DOI：
10.1016/0040-9383(94)90038-8
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发表时间：
1994
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影响因子：
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通讯作者：
S. Kamada
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文献类型：
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作者：
S. Kamada
1990年9月，他在大阪市立大学的一次演讲中说。丫。维罗引入了二维编织的概念。如果由第二因子投影0:x 0:+ Dj导出的映射F—P 0：是0：的m倍分支覆盖，并且F到0:x t3Dj的约束是Df上固定m个点与aDi的乘积，则在0:x 0：中平滑正确嵌入0:x 0：中的紧致定向曲面F称为二维m-辫；在L. Rudolph[7,8]意义上，它也被认为是具有同平凡边界的编织曲面。F在0:x S2= Df x （0: u D2）中被自然地扩展到一个封闭定向曲面F，当我们在R4中将0:x S2与标准2球S2的管状邻域识别时，F在R4中被称为一个封闭的二维m-辫。他提到，在R4中平滑嵌入的每一个封闭定向表面都可以被描述为R4中的一个封闭的二维编织。他的证明思路似乎是将Alexander的论证[1]应用于R3中，该论证给出了一种绕核心z轴缠绕定向连杆的方法，直接应用于R4中以标准S2为核心的封闭定向曲面，但细节尚未公布。我们的定理1给出了他的结果的另一种证明。我们使用运动图像方法（参见[3,61]）证明了这一点，并且仅将Alexander的论证应用于R3中的链路，因此它很容易可视化。事实上，通过我们的方法可以很容易地将R4中由运动图像方法描述的给定表面变形成一个封闭的二维辫子。虽然定理1是通过假设Viro的结果（$3）反过来得到的，但我们在这里用我们的方法从上面的原因证明它。作为定理1的应用，我们给出了一个充要条件（定理2），即对于任意给定的整数c21和g20，群g是嵌入在R4中的闭可定向c分量g曲面的群。这个条件是由某个Wirtinger表示给出的(参见。[lo])与二维编织有关。已知当且仅当R4中有有限Wirtinger表示（参见[9]）时，群是封闭可定向曲面的群。定理2的重要一点是曲面的属可以被指定。F. Gonzalez-Acufia通过将R4中的曲面变形成辫状[4]，对R4中的曲面群独立得出了类似的结果。R4中涉及到定理1的曲面是他的辫状曲面的某种细化。定理的定义和陈述在$1中。我们用正规化定理[6]和马尔可夫定理证明了上文92中的定理1。在1993年，我们观察到了定理1和维罗的a维闭合辫之间的关系。最后在1994年证明了定理2。
IN A lecture given at Osaka City University in September 1990, 0. Ya. Viro introduced the notion of 2-dimensional braid. A compact oriented surface F smoothly and properly embedded in 0: x 0: is called a 2-dimensional m-braid if the map F--P 0: induced from the second factor projection 0: x 0:+ Dj is an m-fold branched covering of 0: and the restriction of F to 0: x t3Dj is the product of fixed m points on Df with aDi; which is also considered a braided surface with identically trivial boundary in the sense of L. Rudolph [7, 8]. F is naturally extended to a closed oriented surface F in 0: x S2= Df x (0: u D2) and, when we identify 0: x S2 with a tubular neighborhood of a standard 2-sphere S2 in R4, F is called a closed 2-dimensional m-braid in R4. He mentioned that every closed oriented surface smoothly embedded in R4 can be described as a closed 2-dimensional braid in R4. The idea of his proof seems to be applying Alexander’s argument [l], which gives a method of winding oriented links in R3 around the core z-axis, directly to closed oriented surfaces in R4 with a standard S2 as the core, but the details are unpublished. Our Theorem 1 gives an alternative proof of his result. We prove it by using the motion picture method (cf.[3, 61) and applying Alexander’s argument only to links in R3, therefore it is easy to visualize. In fact by our method one can easily deform a given surface in R4 described by the motion picture method into a closed 2-dimensional braid. Although Theorem 1 is conversely obtained by assuming Viro’s result ($3), we prove it here by our method from the above reason.As an application of Theorem 1, we give a necessary and sufficient condition (Theorem 2) that for any given integers c 2 1 and g 2 0, a group G is the group of a closed orientable c-component genus g surface embedded in R4. This condition is given by a certain Wirtinger presentation(cf.[lo]) related to the 2-dimensional braid. It is known that a group is the group of a closed orientable surface in R4 if and only if it has a finite Wirtinger presentation (cf.[9]). An important point of Theorem 2 is that the genus of the surface can be specified. F. Gonzalez-Acufia independently has a similar result on groups of surfaces in R4 by deforming surfaces in R4 to be like braids [4]. Our surfaces in R4 involved in Theorem 1 is a certain kind of refinement of his braidlike surface. Definitions and statements of Theorems are in $1. We prove Theorem 1 in 92 as stated above, with help of the normalization theorem [6] and Markov’s theorem. In 93 a relationship between Theorem 1 and Viro’s closed a-dimensional braids is observed. Finally Theorem 2 is proved in 94.