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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2 Answers
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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 :( )
answered Mar 29, 2010 at 19:45
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$\begingroup$ How on Earth do you pronounce that? $\endgroup$ Commented Mar 29, 2010 at 19:53
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1$\begingroup$ I would read "higher cup products". $\endgroup$ Commented Mar 29, 2010 at 19:57
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2$\begingroup$ Judging by the LaTeX, "LOL-i products" maybe? $\endgroup$ Commented Mar 29, 2010 at 21:30
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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
