Self-intersections of immersions and Steenrod operations

We present a formula describing the action of a generalised Steenrod operation of Z(2)-type [14] on the cohomology class represented by a proper self-transverse immersion f: Ma dagger not signX. Our formula depends only on the Umkehr map, the characteristic classes of the normal bundle, and the class represented by the double point immersion of f. This generalises a classical result of R. Thom [13]: If alpha is an element of H (k) (X; Z(2)) is the ordinary cohomology class represented by f: Ma dagger not signX, then Sq (i) (alpha) = f(*) w (i) (nu(f) ).