  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.
