Families of Maps from the Plane to The Plane
Families of Maps from the Plane to The Plane
复制标题
从飞机到飞机的地图系列
DOI：
10.1112/jlms/s2-36.2.351
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发表时间：
1987
影响因子：
1.2
通讯作者：
J. H. Rieger
中科院分区：
文献类型：
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作者：
J. H. Rieger
自惠特尼的经典工作以来，从(实的或复的)平面到平面的映射的奇异性一直是大量研究的主题。虽然Whitney和Mather[12]研究了这些映射的稳定奇点，但最近的工作主要集中在从平面到平面的映射族中包含的低余维退化奇点[1，2，3，7]。这些奇点有一个很好的视觉解释：它们的判别曲线对应于光滑(透明)表面或“明显轮廓”的轮廓。这些结果在图像理解领域的应用已在[5，9]中有所描述.这里我们给出了从平面到平面的余弦一的所有简单奇点的列表(参见.4.1关于简单的定义)。我们还对余维小于等于6的余维为1和j^-余维的所有奇点进行了分类。这些参数包含在垂直四参数族中。我们的分类过程包括喷流空间的j^-不变分层和正规形式的(精确)确定度的计算。对于所有的范式，我们计算了一组与它们的判别式和临界集相关的几何stf不变量，它们都是平面曲线。这里，我们照常定义临界集GB的局部重数m/0)和//-和(5-不变量(见[13])。复解析映射芽的不变量c(F)和d(F)是集中在原点的尖点和横折交叉点的个数。(当然，c(F)和d(F)对于实细菌i没有这样的几何解释，尽管在这种情况下它们是尖点和折叠交叉点数量的上界。)我们的分类包含两个无穷级数的简单芽，它们的判别式具有关于c(J)和d(F)的几何描述。然而，并不是我们分类的所有细菌都可以通过这些不变量来区分。这并不奇怪，因为尽管Gaffney和Du Plessis[6，17]的一般结果表明，好的映射芽的特征在于它们的判别式，但目标维度为2的映射子通常不是好的。(例如，我们的分类包含判别式是微分同胚但属于不同的j/类的映射芽。)下面的定理总结了我们的主要结果。
The singularities of maps from the (real or complex) plane to the plane have been the subject of a great number of investigations ever since the classical work of Whitney [19]. Whereas Whitney and Mather [12] have studied the stable singularities of these maps, more recent work has concentrated on degenerate singularities of low codimension contained in families of maps from the plane to the plane [1, 2, 3, 7]. These singularities have a nice visual interpretation: their discriminant curves correspond to the profiles of smooth (transparent) surfaces or'apparent contours'. Applications of these results in the area of image understanding have been described in [5, 9].Here we present a list of all simple singularities of corank one from the plane to the plane (cf. 4.1 for the definition of simple). We also classify all singularities of corank one and j^-codimension less than or equal to six. These are contained in versal 4-parameter families. Our classification procedure consists of an j^-invariant stratification of the jet-space and the calculation of the (exact) determinacy degree of the normal forms. For all normal forms we calculate a set of geometrical stf-invariants associated with their discriminants and their critical sets, which are both plane curves. Here we define the local multiplicity m/0) and the//-and (5-invariants of the critical set£ as usual (see [13]). The invariants c (f) and d (f) of a complexanalytic map-germ/are the number of cusps and transverse fold crossings concentrated at the origin (cf. Section 4).(Of course, c (f) and d (f) have no such geometrical interpretation for real germs/, although they are upper bounds on the number of cusps and fold crossings in this case.) Our classification contains two infinite series of simple germs, whose discriminants have geometrical descriptions in terms of c (J) and d (f). However, not all the germs of our classification can be distinguished by these invariants. This is not too surprising, since although general results of Gaffney and du Plessis [6, 17] imply that good map-germs are characterised by their discriminants, maps of target dimension two are not in general good.(For example, our classification contains map-germs whose discriminants are diffeomorphic but belong to different j/-classes.) The following theorem summarises our main results.