Stiefel—Whitney currents

Abstract

A canonically defined mod 2 linear dependency current is associated to each collection v of sections, v₁,…,v_m, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v 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 d-closed locally integrally flat current of degree q = n −m + 1 and hence determines a ℤ₂-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-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 v. The mod 2 reduction of this current is the mod 2 linear dependency current. The cohomology class of the linear dependency current is 2-torsion and is the qth 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.