Write $H^*(\mathbb{R}P^\infty; \mathbb{Z}_2) = \mathbb{Z}_2[\alpha]$, $\deg \alpha = 1$. What is $Sq^i(\alpha^j)$ for all $i$ and $j$? I am not an algebraic topologist by trade but need to know this result...
1 Answer
$\begingroup$
$\endgroup$
The Cartan formula $$Sq^i(xy)=\Sigma _{j+k=i}Sq^j(x)Sq^k(y)$$ together with the instability condition $$Sq^d(\alpha)=\alpha ^2 \mbox{ if $d=deg(\alpha )$}, Sq^i(\alpha )=0 \mbox{ if $d>deg(\alpha )$}$$ and the property $Sq^0=id$ (see, e.g. https://en.wikipedia.org/wiki/Steenrod_algebra) give $Sq^i (\alpha ^j)= \binom{j}{i}\alpha ^{i+j} $. All of these can be found, for example, http://isites.harvard.edu/fs/docs/icb.topic1525101.files/steenrod-operations.pdf page 8.
