The geometric Hopf invariant and double points

TL;DR

This paper relates the geometric Hopf invariant of a stable map to the double point set of an immersion, providing a new perspective on the classification of immersions in certain dimensions.

Contribution

It expresses the geometric Hopf invariant of the Umkehr map in terms of the immersion's double points and interprets the regular homotopy classification via this invariant.

Findings

The geometric Hopf invariant can be described using double point data.

The classification of immersions in the metastable range is linked to the Hopf invariant.

The approach offers a new algebraic-topological perspective on immersion theory.

Abstract

The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in terms of the double point set of f. We interpret the Smale-Hirsch-Haefliger regular homotopy classification of immersions f in the metastable dimension range 3m<2n-1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.