TITLE: Why are cup-i products and Steenrod Squares often (always?) unary?
QUESTION [5 upvotes]: One way to define the Steenrod Operations is to use the cup-i product, as in Mosher and Tangora's book.  It basically says, given the chain complex from mod-2 homology $C_\ast$, define
$D_0 : C_\ast\to C_\ast\otimes C_\ast$, 
so that the cup product is given (on cocycles)
$(u\cup v)(\sigma) = (u\otimes v)(D_0\sigma)$
and then for higher $i$, define $D_i$ so that 
$D_{i-1}+\rho D_{i-1} = D_i\partial + \partial D_i$
where $\rho$ is the flipping map.  Then the $\cup_i$ product is just
$(u\cup_i u)(\sigma) = (u\otimes u)(D_i\sigma)$
And then define for $[u]\in H^n$
$Sq^{2n-i}([u]) = [u\cup_{i}u]$
This definition seems perfectly well-defined as a binary operation, and yet wherever I've seen it done it has only even been used as a unary operation.  
Is there a reason why this is the case, why either the product is undefined as a binary product or not useful as a binary product or just too hard to use?
Is this a dumb question?
Thanks,
-Joseph

REPLY [8 votes]: Under suitable hypotheses, Gugenheim and I use the binary $\cup_1$ and especially the fact that it is a graded derivation (Hirsch formula) as the key to giving a calculation of $H^*(G/H)$ as the torsion product over $H^*(BG)$ of $R$ and $H^*(BH)$, where coefficients are taken in a suitable commutative ring $R$.  This even works for suitable $H$-spaces.   See
http://www.math.uchicago.edu/~may/BOOKS/GugMay.pdf and 
http://www.math.uchicago.edu/~may/PAPERS/MNApril20.pdf