到Stiefel流形的映射与交叉帽奇点
Mappings into the Stiefel manifold and cross-cap singularities
摘要：
取n>k>1，使得n-k是奇数。本文考虑将（n-k+1）维闭球映射到（n \times k）-矩阵空间，使得其对球面的限制进入Stiefel流形V_k（R^n）。我们构造了一个A的同伦不变量Λ| S^{n-k}定义了V_k（\R^n）的同伦的第（n-k）个群与Z_2之间的同构。它可以用来计算在一个有效的方式类|S^{n-k}在这个同伦群中的一个多项式映射a，并求从一个m维闭球到R^{2 m-1}（m为偶数）的映射的交叉帽奇点的模2个数。
英文摘要：
Take n>k>1 such that n-k is odd. In this paper we consider mapping a from (n-k+1)-dimensional closed ball into the space of (n \times k)--matrices such that its restriction to a sphere goes into the Stiefel manifold V_k(R^n). We construct a homotopy invariant Λ of a|S^{n-k} which defines an isomorphism between (n-k)-th group of homotopy of V_k(\R^n) and Z_2. It can be used to calculate in an effective way the class of a|S^{n-k} in this homotopy group for a polynomial mapping a and to find the number mod 2 of cross-cap singularities of a mapping from a closed m-dimensional ball into R^{2m-1}, m even.
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