# Homotopy commutativity of the cup product

Where can I find explicit formulas for the higher homotopies, which exhibit the cup product (in singular simplicial cohomology, say) as homotopy commutative on the cochain level? Same question in Cech cohomology.

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Such homotopies are given by the $\smile_i$-products. Steenrod gives explicit formulas, IIRC, in [Steenrod, N. E. Products of cocycles and extensions of mappings. Ann. of Math. (2) 48, (1947). 290--320. MR0022071], but the easiest is to prove they exist using acyclic models.

(Maybe Steenrod only deals with $\mathbb Z_2$ coefficients? I don't have access to the paper now :( )

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How on Earth do you pronounce that? –  Qiaochu Yuan Mar 29 '10 at 19:53
I would read "higher cup products". –  Mariano Suárez-Alvarez Mar 29 '10 at 19:57
Or "cup-i products". –  Charles Rezk Mar 29 '10 at 20:15
Judging by the LaTeX, "LOL-i products" maybe? –  darij grinberg Mar 29 '10 at 21:30
These operations in the singular setting were fully and explicitly developed and generalized beautifully by McClure and Smith (who also credit Benson and Milgram) in their paper "Multivariable cochain operations and little $n$-cubes": http://arxiv.org/pdf/math.QA/0106024.pdf