# Geometric interpretation of the conjugation operation in the dual Steenrod algebra

As the dual mod 2 Steenrod algebra, $A$, is a Hopf algebra, it has the conjugation operation, $\chi:A\to A$. Milnor also gives a formula for this.

I wonder if there is any source telling about a geometric interpretation of this operation.

• What do you mean by geometric? You can identify $A$ with $\pi_*(H\wedge H)$, and $\chi$ comes from the map that switches the two factors of $H$. That is a spectrum-level description, which might or might not be considered geometric. Oct 3 '15 at 15:32
• You can also interpret $\chi$ as part of the structure of an affine group scheme. Is that what you mean by geometric? Oct 5 '15 at 11:41

I do not know if this is what you are searching for, but for the Steenrod algebra itself $\chi$ is closely related to Spanier-Whitehead duality. Let $X$ be a spectrum and $DX=F(X,S)$ 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 $\chi \Theta$.
When $X=M$ is a smooth manifold, Atiyah duality identifies $DM$ with the Thom spectrum of $Th(\nu)$ where $\nu=-TM$ is the stable normal bundle of $M$. And we get an isomorphism
$$H_i(M,\mathbb{F}_2)\cong Hom(H^i(M,\mathbb{F}_2),\mathbb{F}_2)\rightarrow H^{-i}(DM,\mathbb{F}_2)\cong H^{dim(M)-i}(M,\mathbb{F}_2).$$
This explains the Wu formula, that relates the total Stiefel-Whitney classe $w(M)$ of $M$ with the Wu class $v(M)$: $$v(M)=\chi(Sq)(w(M)).$$