Return to Revisions

I do not know if this is what you are searching for, but this is closely related to Spanier-Whitehead duality. Let $X$ be a spectrum and $DX$ its Spanier-Whitehead dual we have an isomorphism $$Hom(H^*(X,\mathbb{F}_2),\mathbb{F}_2)\rightarrow H^{-*}(DX,\mathbb{F}_2)$$ the contragredient action on the left of a Steenrod operation $\Theta$ corresponds to the action on the right of $\chi \Theta$.