The author defines a locally Lebesgue integrable current, called the Stiefel-Whitney spark, as follows. Let \(\alpha\) be a collection of \(m\) sections, verifying a weak measure-theoretic condition called “\(LD\)-atomicity”, of a rank \(n\) real vector bundle, \(F\to X\), with metric connection and suppose that \(q=n-m+1\) is odd. The \(q\)-th Stiefel-Whitney spark \(S_q (\alpha)\) of \(\alpha\) is a locally integrable form which satisfies the fundamental current equation \(dS_q(\alpha)=-LD(\alpha)\) on \(X\), where \(LD (\alpha)\) is the linear dependency current of the collection of sections \(\alpha\) defined by F. R. Harvey and the author in [J. Geom. Anal. 8, No. 5, 809-844 (1998; Zbl 1054.53088)]. (In the quoted paper has been proved that the integer cohomology class of \(LD(\alpha)\) is the \(q\)th integer Stiefel-Whitney class \(W_q (F)\) of \(F\to X)\).
The main results of the paper are an explicit local formula for the Stiefel-Whitney spark \(S_q(\alpha)\) and that this current verifies the local current verifies the local current equation \(2S_q (\alpha)-R= dL\) on \(X\), where \(L\) is \(L^1_{\text{loc}}\) and \(R\) is a locally rectifiable current whose mod 2 reduction represents the \((q-1)\)st mod 2 Stiefel-Whitney class \(w_{q-1}\in H^{q-1} (X;\mathbb{Z}_2)\).
Moreover, the author proves that the Stiefel-Whitney spark yields a natural generalization of Eells’ method [J. Eells jun., Trans. Am. Math. Soc. 92, 142-153 (1959; Zbl 0088.38003)] of representing Stiefel-Whitney classes by pairs of forms with singularities.