The geometric Hopf invariant and double points

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 -> 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.