Let $x\in (H\mathbb F_2)_n(X)=[S^n,H\mathbb F_2 \wedge X]$ be a homology class for a space $X$. Is there a description of $$[S^n\overset x\to H\mathbb F_2 \wedge X\overset{Sq^r\wedge id}\to \Sigma^r H\mathbb F_2\wedge X]\in (H\mathbb F_2)_{n-r}(X)$$ in terms of Steenrod operations and the Kronecker pairing?
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2$\begingroup$ <xSq^r, u>=<x,Sq^ru> determines it (where you’ll have to trace through some conventions to figure out if the antipode shows up when moving the square over) $\endgroup$– Dylan WilsonCommented Oct 3, 2019 at 12:08
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$\begingroup$ Than you Dylan. I just found a reference for your statement in On Thom Spectra, Orientability, and Cobordism by Rudyak, Yu. B. II.6.36. $\endgroup$– syzygCommented Oct 3, 2019 at 13:31
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