The Stiefel-Whitney spark

In this paper de Rham currents and geometric measure theory are used to study the mod 2 and integer Stiefel-Whitney classes. The paper is part of a larger program of Harvey and his collaborators to develop a theory of primary and secondary characteristic currents. In a previous paper we proved that, in the case of the integer Stiefel-Whitney classes, to each "atomic" collection of sections of a real vector bundle there is associated a linear dependency current, with rectifiability properties, which is supported on the linear dependency set of the collection. The integer cohomology class of a linear dependency current is an integer Stiefel-Whitney class. In this paper a locally Lebesgue integrable current, called the Stiefel-Whitney spark, is associated to each atomic collection of sections. The Stiefel-Whitney spark satisfies the local current equation that its exterior derivative is equal to the linear dependency current. This equation is the natural analogue for the integer Stiefel-Whitney classes of Harvey and Lawson's local Gauss-Bonnet-Chern formula for the Euler class. (A similar current equation is derived for the mod 2 Stiefel-Whitney classes.) Consequently, the Stiefel-Whitney spark plays the same role for the Stiefel-Whitney class that Chern's spherical potential (or transgression) plays for the Euler class. An explicit local formula is derived for the Stiefel-Whitney spark, analogous to Chern's formula for the spherical potential. Furthermore, the Stiefel-Whitney spark yields a natural generalization of Eells' method of representing Stiefel-Whitney classes by pairs of forms with singularities.