TITLE: Homotopy commutativity of the cup product
QUESTION [13 upvotes]: 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.

REPLY [12 votes]: 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