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    Graduate Cubical cup poduct with a unit 1-cochain and the coboundary operator

    Define a u-coboundary operator u-d on a cubic p-cochain f as cup product multiplication U from the left with a unit 1-cochain: (u-d)f = 1Uf. Because 1U1=0 and associativity of U, we have (u-d)^2=0. What is the relation of u-d and the standard coboundary operator d? Are they the same?