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Generalisation of Hirsch formula for the associativity of Steenrod's higher $\cup_2$ product...

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 …