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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 suchthat its restriction to a sphere goes into the Stiefel manifold V_k(R^n). Weconstruct a homotopy invariant \Lambda\ of a|S^{n-k} which defines anisomorphism between (n-k)-th group of homotopy of V_k(\R^n) and Z_2. It can beused to calculate in an effective way the class of a|S^{n-k} in this homotopygroup for a polynomial mapping a and to find the number mod 2 of cross-capsingularities of a mapping from a closed m-dimensional ball into R^{2m-1}, meven.

Mappings into the Stiefel manifold and cross-cap singularities