Stiefel-Whitney Currents

TL;DR

This paper introduces a new mod 2 linear dependency current associated with sections of a real vector bundle, which generalizes previous concepts and relates to Stiefel-Whitney classes, with applications to singularities and degeneracies.

Contribution

It defines a canonical mod 2 dependency current for vector bundle sections that extends general position results and connects to Stiefel-Whitney classes, including twisted and higher degeneracy currents.

Findings

The dependency current is mod 2, closed, and determines the Stiefel-Whitney class.

The current is well-defined and independent of section choice.

Applications include analysis of singularities and degeneracies in bundle maps.

Abstract

A canonically defined mod 2 linear dependency current is associated to each collection of m sections of a real rank n vector bundle. This current is supported on the linear dependency set of the collection of sections. It is defined whenever the collection satisfies a weak measure theoretic condition called "atomicity". Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2, closed, locally integrally flat current of degree q= n-m+1 and hence determines a mod 2 cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover it is the q-th Stielfel-Whitney class of the vector bundle. More is true if q is odd or q=n. In this case a linear dependency current which is twisted by the orientation of the bundle can be associated to the collection. The cohomology class of the linear dependency current is 2-torsion and is the q-th twisted integral Stiefel-Whitney class of the bundle. In addition, higher dependency and general degeneracy currents of bundle maps are studied, together with applications to singularities of projections and maps. These results rely on a theorem of Federer which states that the complex of integrally flat currents mod p computes cohomology mod p. An alternate approach to Federer's theorem is offered in an appendix. This approach is simpler and is via sheaf theory.