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

## TL;DR

This paper extends the concept of intersection number to stratified sets like matrices of positive corank, linking it to homotopy invariants and providing methods for computation and applications in counting singularities.

## Contribution

It introduces an extension of intersection number to stratified sets such as positive corank matrices and connects it with homotopy invariants, offering computational techniques.

## Key findings

- Intersection number coincides with a homotopy invariant for maps into matrices with Sigma.
- Effective methods are provided for computing the intersection number in polynomial cases.
- Applications include counting cross-cap singularities mod 2 or algebraic sum.

## Abstract

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 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 x k) real matrices with Sigma coincides with a homotopy invariant associated with a map going to the Stiefel manifold. In a polynomial case, we present an effective method to compute this intersection number. We also show how to use it to count the number mod 2 or the algebraic sum of cross--cap singularities of a map from an m--dimensional manifold with boundary to R^{2m-1}.