Cross-cap singularities counted with sign

TL;DR

This paper introduces a method to compute the algebraic count of cross-cap singularities in mappings from odd-dimensional manifolds into Euclidean space, with applications to intersection numbers of immersions.

Contribution

It presents a novel approach for counting cross-cap singularities algebraically and relates this to intersection numbers of immersions, expanding understanding of singularity theory.

Findings

Method for computing algebraic number of cross-caps

Application to intersection number of sphere immersions

Provides a link between singularities and intersection theory

Abstract

There is presented a method for computing the algebraic number of cross-cap singularities for mapping from m dimensional compact manifold with boundary M into R^(2m-1), m is odd. As an application, the intersection number of an immersion g from (m-1)-dimensional sphere to R^(2m-2) is described as the algebraic number of cross-caps of a mapping naturally associated with g.