Definition
In singular cohomology, the cup product is a construction giving a product on the graded cohomology ring H∗(X) of a topological space X.
The construction starts with a product of cochains: if cp is a p-cochain and dq is a q-cochain, then
where σ is a (p + q) -singular simplex and is the canonical embedding of the simplex spanned by S into the -standard simplex.
Informally, is the p-th front face and is the q-th back face of σ, respectively.
The coboundary of the cup product of cocycles cp and dq is given by
The cup product of two cocycles is again a cocycle, and the product of a coboundary with a cocycle (in either order) is a coboundary. Thus, the cup product operation passes to cohomology, defining a bilinear operation
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