Self-intersections of Immersions and Steenrod Operations

We present a formula describing the action of a generalised Steenrod operation of $\Z_2$-type on the cohomology class represented by a proper self-transverse immersion $f\co M\imm X$, in terms of the equivariant double points of $f$ and the characteristic classes of its normal bundle. This generalises a classical result of R.\ Thom: If $\alpha\in H^k(X;\Z_2)$ is the ordinary cohomology class represented by $f\co M\imm X$, then $\mathrm{Sq}^i(\alpha)=f_* w_i(\nu_f)$.