Mappings into the Stiefel manifold and cross-cap singularities

2015/07/17 by Iwona Krzyżanowska, Krzyżanowska, Iwona, Aleksandra Nowel +1
Computer Science · Mathematics · #12Y05 #14P25 #57R45 #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1507.04892

openalex publication_date 2015/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 × k)--matrices such that its restriction to a sphere goes into the Stiefel manifold V_k(Rⁿ). 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(\Rⁿ) and Z₂. 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.

Citations

• The general type of singularity of a set of 2n−1 smooth functions of n variables
• Cross-cap singularities counted with sign

Related

• Polynomial mappings into a Stiefel manifold and immersions
• Cross-cap singularities counted with sign
• Intersection number of a map with the set of matrices of positive corank
• p-local decompositions of projective Stiefel manifolds
• Some Observations on projective Stiefel Manifolds