GEOMETRIC FORMULAS FOR SMALE INVARIANTS OF CODIMENSION TWO IMMERSIONS
GEOMETRIC FORMULAS FOR SMALE INVARIANTS OF CODIMENSION TWO IMMERSIONS
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二次浸入余维小不变量的几何公式
DOI：
10.1016/s0040-9383(02)00005-8
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
A. Szűcs
中科院分区：
文献类型：
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作者：
T. Ekholm;A. Szűcs
我们给出了表示(4k−1)-球面到(4k+1)-空间的浸入f的三个公式。公式的项是任意有向4k维流形的任意广义光滑映射g的几何特征，其中限制在边界上的g是f在(6k−1)-空间中的正则同伦。公式表明，如果f和g是(4k−1)-球面到(4k+1)-空间的两个非正则同伦浸没，那么它们也是非正则同伦到(6k−1)-空间的浸入.此外，连接f到g的(6k−1)-空间中的任何泛同伦必至少有Ak(2k−1)！其中，如果k是奇数，则Ak=2，如果k是偶数，则Ak=1。
We give three formulas expressing the Smale invariant of an immersion f of a (4k−1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k−1)-space. The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k−1)-sphere into (4k+1)-space then they are also non-regularly homotopic as immersions into (6k−1)-space. Moreover, any generic homotopy in (6k−1)-space connecting f to g must have at least ak(2k−1)! cusps, where ak=2 if k is odd and ak=1 if k is even.