Intersection number of a map with the set of matrices of positive corank

The definition of the intersection number of a map with a closed manifold can be extended to the case of a closed stratified set such that the difference between the dimensions of its two biggest strata is greater than $1$. The set $\Sigma $ of matrices of positive corank is an example of such a set. It turns out that the intersection number of a map from an $(n-k+1)$-dimensional manifold with boundary into the set of $n\times k$ real matrices with $\Sigma $ coincides with a homotopy invariant associated with a map going to the Stiefel manifold $\widetilde{V}_k(\mathbb {R}^n)$. In the polynomial case, we present an efficient method to compute this intersection number. We also show how to use it to count the mod $2$ number and the algebraic sum of the cross-cap singularities of a map from an $m$-dimensional manifold with boundary to $\mathbb {R}^{2m-1}$.