Generalisation of Hirsch formula for the associativity of Steenrod's higher $\cup_2$ product with $\cup_1$ and cup products

Question

For $f$, $g$ and $h$ cochains, the Hirsch formula is given as $$ (f\cup g)\cup_1 h=f\cup (g\cup_1 h)+(-1)^{q(r-1)}(f\cup_1 h)\cup g.$$ Is there a more general formula that relates the associativity of $\cup_2$ product with $\cup_1$ and $\cup$?

For example, is it known how to express $f\cup (g\cup_2 h)$ in terms of $(f\cup g)\cup_2 h$ plus some other terms or $f\cup_1 (g\cup_2 h)$ in terms of $(f\cup_1 g)\cup_2 h$ or something else that involves $\cup_2$, $\cup_1$ and $\cup$? If it is not known, what is a quick method to work this out?

Answer 1

If I may be allowed to interpret loosely what you're trying to do, maybe it's to understand $E_\infty$ structures in a way that's comparable to the very algebraic $A_\infty$ story, or at the very least the low degree parts of them. If so, the story is much more complicated, but it can be done. This is in terms of the "surjection operad" described in Section 2 of McClure and Smith, "Multivariable cochain operations and little $n$-cubes". They explicitly acknowledge that although I didn't know what an operad was in 1991 when I wrote "Representations and Cohomology II", my algebraic formulation of the Steenrod operations in Section 4.5 there is basically what is reflected in their operad. I hope you can answer your question by looking at the low degree parts of this operad. For $p=2$ it's a lot simpler than for $p$ odd. You might also want to chase up on MathSciNet the papers that refer to the McClure-Smith paper.