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Cup-i products extend cup products.
They can be used to define Steenrod squares in the same manner as ordinary cup products can be used to define the square of a cohomology class.
Given a simplicial set X and i≥0, we define the cup-i product as the map on simplicial cochains on X with coefficients in Z/2 induced by the map on simplicial chains
Δi:C(X,Z/2)→C(X,Z/2)⊗C(X,Z/2),where Δi evaluate on an n-simplex x∈Xn is 0 if iUnknown charactern and
∑UdU0(x)⊗dU1(x),where U⊂{0,…,n} has cardinality n−i and
Uk={u∈U∣r(u)+k=u(mod2)},where r(u)=#{v∈U∣v≤u}.
Anibal M. Medina-Mardones, New formulas for cup-i products and fast computation of Steenrod squares, arXiv.
Ralph M. Kaufmann, Anibal M. Medina-Mardones, Cochain level May-Steenrod operations, arXiv.
Anibal M. Medina-Mardones, An axiomatic characterization of Steenrod’s cup-i products, arXiv.
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