Mappings into the Stiefel manifold and cross-cap singularities

TL;DR

This paper develops a homotopy invariant to classify polynomial mappings into the Stiefel manifold and determine the number of cross-cap singularities in mappings from balls into Euclidean space.

Contribution

It introduces a new homotopy invariant  that links homotopy groups of the Stiefel manifold to , enabling effective classification and singularity counting for polynomial maps.

Findings

Defines a homotopy invariant  for mappings into the Stiefel manifold.

Establishes an isomorphism between homotopy groups and Z_2 using .

Provides a method to count cross-cap singularities mod 2 for polynomial maps.

Abstract

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 \Lambda\ 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.