Antipodal vector bundle monomorphisms

Abstract

In this paper we study existence and classification questions concerning antipodal vector bundle monomorphismsu (i.e.,u is regularly homotopic to its negative −u). In a metastable dimension range the singularity approach yields complete obstructions which, however, have to be weakened usually in order to become computable. In many situations we determine the resulting “weak, stable” invariants completely; a central role here is played by the antipodality obstructionv i(α, β), a curious combination of Stiefel-Whitney classes. Moreover, in some sample cases we describe precisely how much information gets lost by the transition to these weaker invariants. This involves, e.g., identifying some classical second order obstructions. As an application we exhibit a setting where the difference invariantd(u, −u) distinguishes all (and in fact, infinitely many) regular homotopy classes. Also, we give complete existence and enumeration results for nonstable and stable tangent plane fields on complex projective spaces in terms of explicit numerical conditions.