A Formula for Deviation from Commutativity: The Transfer and Steenrod Squares

Abstract

The ordinary cohomology transfer associated to the orbit space projection of a finite group action need not commute with stable cohomology operations. In particular, if an even group acts on a space, the resulting transfer $\tau$ will not generally commute with the Steenrod squares, ${\text {S}}{{\text {q}}^i}$. This paper contains a formula for the deviation from commutativity $({\text {S}}{{\text {q}}^i}\tau - \tau {\text {S}}{{\text {q}}^i})x$ in the case of an involution. The formula involves the restriction of $x$ to the cohomology of the fixed point set, as well as certain naturally occurring characteristic classes.