Desingularizing maps of corank one
Desingularizing maps of corank one
复制标题
科兰克一号地图的去奇异化
DOI：
10.1090/s0002-9939-1980-0581010-8
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发表时间：
1980
期刊：
影响因子：
--
通讯作者：
C. Curley
中科院分区：
文献类型：
--
作者：
S. J. Blank;C. Curley
1960年，A. Haefliger提出的必要和充分条件因素的顺利地图从表面到平面通过沉浸到R3。本文给出了n维流形之间的映射通过浸入值域上的线丛而分解的充要条件，其中n > 2。此外，条件的因式分解这样的地图通过淹没从线丛域。通过流形之间的光滑映射f：M -* V的去奇异化，我们指的是以下之一。(1)浸没F：对于某个线束m：E -> V，M -> E，使得f = m ° F，或者（2）对于具有零截面s的某个线束m '：E' -> M，浸没F '：E' -> F，使得f = F' ° s。我们说/ desingularizes在第一（resp。第二，相对于E（resp.£”）。如果f在第一个意义上具有去奇异化，那么法向束到F的像的拉回提供了一个线束，关于该线束f在第二个意义上具有去奇异化。匡威则不然。图1的映射在第二种意义上去奇异化，但不是第一种意义上去奇异化。在本文中，我们提出的必要和充分条件，去奇异的一般映射关于一个给定的丛。在[1] A. Haefliger提供了通过浸入R3将一般映射从闭合曲面分解到平面的充分必要条件。令/：M -> V是一个光滑映射，其中M是一个封闭的光滑n-流形，F是一个光滑的n-流形，令m：E-> V是一个线丛。如果存在浸入F：M -» E，其中f = m ° F，我们说f通过浸入E而分解。如果f是因子，则处处秩为df > n - 1，或等价的，f的corank不超过1。一般f的奇性，我们指的是2-一般的和有限到一的[2]，处处具有秩df > n - 1，由流形δ，= {x G A/|秩dfx = n [I]，S1，i =（x G5，|秩d（fs）x = n2}.映射f是2-类属的，保证了Sx和Sxx是M的子流形。如果我们把Klein瓶看作{（</>，0）|<p G Sx，0 E [0，2w]} mod关系（</>，0）=（-<p，2w）则f（<p，9）=（cos <p + 2，f？）是从克莱因瓶到平面的映射（极坐标）。我们将证明这个映射在第二种意义上去奇异化，但在第一种意义上不去奇异化。图1显示了该地图的图像。编辑于1979年6月21日收到，修订版于1979年9月29日收到。AMS（MOS）主题分类（1970年）。小学57 D35、57 D40;中学57 D20、57 D45。
In 1960, A. Haefliger presented necessary and sufficient conditions for factoring a smooth map from a surface into the plane through an immersion into R3. Here, necessary and sufficient conditions are given for factoring a map between manifolds of dimension n, n > 2, through an immersion into a line bundle over the range. In addition, conditions are given for factoring such a map through a submersion from a line bundle over the domain. By a desingularization of a smooth map f: M -* V between manifolds we mean one of the following. (1) An immersion F: M —> E for some line bundle m: E -» V such that/ = m ° F or (2) a submersion F': E' -> F for some line bundle m': E' —» M with zero section s such that/ = F' ° s. We say / desingularizes in the first (resp. second) sense with respect to E (resp. £"). If/has a desingularization in the first sense, then the pull-back of the normal bundle to the image of F provides a line bundle with respect to which / has a desingularization in the second sense. The converse is not true. The map for Figure 1 desingularizes in the second sense but not the first. In this paper we present necessary and sufficient conditions for desingularizing a generic map with respect to a given bundle. In [1] A. Haefliger provides necessary and sufficient conditions for factoring a generic map from a closed surface into the plane through an immersion into R3. Let /: M —> V be a smooth map where M is a closed smooth n-manifold and F is a smooth n-manifold and let m: £—> V be a line bundle. If there is an immersion F: M —» E with / = m ° F we say / factors through an immersion into E. If / does factor then rank df > n — 1 everywhere, or equivalent^, the corank of / does not exceed one. The singularities of a generic /, by which we mean 2-generic and finite-to-one [2], with rank df > n — 1 everywhere, are stratified by the manifolds 5, = {x G A/|rank dfx = n I], S,, = (x G 5,|rank d(f¡s)x = n 2}. The map/being 2-generic ensures that Sx and Sxx are submanifolds of M. If we regard the Klein bottle as {(</>, 0)|<p G Sx, 0 E [0, 2w]} mod the relation (</>, 0) = (-<p, 2w) then /(<p, 9) = (cos <p + 2, f?) is a map from the Klein bottle to the plane (in polar coordinates). We will show that this map desingularizes in the second sense but not in the first sense. Figure 1 represents the image of this map. Received by the editors June 21, 1979 and, in revised form, September 29, 1979. AMS (MOS) subject classifications (1970). Primary 57D35, 57D40; Secondary 57D20, 57D45.