A generalization of Banchoff’s triple point theorem
A generalization of Banchoff’s triple point theorem
复制标题
班霍夫三点定理的推广
DOI：
10.1090/s0002-9939-98-04083-0
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
A. Szűcs
中科院分区：
文献类型：
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作者：
P. Akhmetiev;Richárd Rimányi;A. Szűcs
考虑将表面浸入 S3 中。班乔夫定理指出，三相点数的宇称性与表面的欧拉特性的宇称性是一致的。在这里，我们将该定理推广到球体中任意偶维流形的余维 1 浸没。该证明类似于 1977 年 R. Fenn 和 P. Taylor 以预印本形式传播的班乔夫定理的证明。让我们考虑球 S 中闭流形 M 的余维 1 平滑泛型（即自横向）浸没 f。让我们回想一下 i 元组点（在 R ⊂ S 中）的邻域在这种自横向浸没中的样子。考虑 R 中的坐标超平面，并取该配置与 Rn+1−i 的直积。所获得的结果与 f 图像中 i 元组点的邻域微分同胚。对于任何自然数 i，1 ≤ i ≤ n+ 1，让我们用 Δi 表示 S 中 i 元组点的集合，即 Δi = {y ∈ S | f−1(y) 由 i 个不同的点组成}。众所周知，dim Δi = n + 1 − i，且 ⋃∞ r=i Δr 是浸没流形（尽管它不是一般位置，即它是非自横向浸没的图像）。令 Δi 为闭流形，使得 ⋃∞ r=i Δr 为 Δi 在 S 中的浸没图像。评论。当然，许多不同的流形可以浸入S中，使得它们的图像为⋃∞ r=i Δr。例如，如果给定一个可能的 Δi，那么它的任何有限覆盖也可以发挥作用。我们通过假设 f 的 i 元组点是浸没 Δi # S 的非重数点来明确选择 Δi。我们将流形 Δi 称为 f 的 i 元组流形。我们的定理声称，对于 n，i 元流形的欧拉特征之和是偶数。 （对于 n = 2，这正是班乔夫定理。）编辑于 1995 年 7 月 4 日收到，并于 1996 年 9 月 2 日修订。1991 年数学学科分类。主要 57R42。
Consider an immersion of a surface into S3. Banchoff’s theorem states that the parity of the number of triple points and the parity of the Euler characteristic of the surface coincide. Here we generalize this theorem to codimension 1 immersions of arbitrary even dimensional manifolds in spheres. The proof is an analogue of a proof of Banchoff’s theorem circulated in preprint form due to R. Fenn and P. Taylor in 1977. Let us consider a codimension 1 smooth generic (i.e. self-transverse) immersion f of a closed manifold M in the sphere S. Let us recall how a neighborhood of an i-tuple point (in R ⊂ S) looks like in such a self-transverse immersion. Consider the coordinate hyperplanes in R and take the direct product of this configuration with Rn+1−i. What is obtained is diffeomorphic to the neighborhood of an i-tuple point in the image of f . For any natural number i, 1 ≤ i ≤ n+ 1, let us denote by ∆i the set of i-tuple points in S, i.e. ∆i = {y ∈ S | f−1(y) consists of i different points}. As is well known, dim ∆i = n + 1 − i, and ⋃∞ r=i ∆r is an immersed manifold (although it is not in general position, i.e. it is the image of a non-self-transverse immersion). Let ∆i be a closed manifold such that ⋃∞ r=i ∆r is the image of an immersion of ∆i in S . Remark. Of course, many different manifolds can be immersed into S so that their images are ⋃∞ r=i ∆r. For example if a possible ∆i is given, then any of its finite coverings serves as well. We make the choice of ∆i explicit by assuming that the i-tuple points of f are non-multiple points of the immersion ∆i # S. We shall call the manifold ∆i the i-tuple manifold of f . Our theorem claims that for n even the sum of the Euler characteristics of i-tuple manifolds is even. (For n = 2 this is exactly Banchoff’s theorem.) Received by the editors July 4, 1995 and, in revised form, September 2, 1996. 1991 Mathematics Subject Classification. Primary 57R42.