Self-intersections of Immersions and Steenrod Operations

We present a formula describing the action of a generalised Steenrod operation of ${\mathbb{Z}}_2$-type on the cohomology class represented by a proper self-transverse immersion $f\text{co}M\text{imm}X$, 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 $\alpha \in H^k(X;{\mathbb{Z}}_2)$ is the ordinary cohomology class represented by $f\text{co}M\text{imm}X$, then ${\mathrm{S}\mathrm{q}}^i(\alpha )=f_{\ast }{w}_i({\nu }_f)$.