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Concept
Cup product
Summary
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H∗(X), called the cohomology ring. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944.
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 is a p-cochain and
is a q-cochain, then
where σ is a singular (p + q) -simplex and
is the canonical embedding of the simplex spanned by S into the -simplex whose vertices are indexed by .
Informally, is the p-th front face and is the q-th back face of σ, respectively.
The coboundary of the cup product of cochains and 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. The cup product operation induces a bilinear operation on cohomology,
The cup product operation in cohomology satisfies the identity
so that the corresponding multiplication is graded-commutative.
The cup product is functorial, in the following sense: if
is a continuous function, and
is the induced homomorphism in cohomology, then
for all classes α, β in H *(Y). In other words, f * is a (graded) ring homomorphism.
It is possible to view the cup product as induced from the following composition:in terms of the chain complexes of and , where the first map is the Künneth map and the second is the map induced by the diagonal .
This composition passes to the quotient to give a well-defined map in terms of cohomology, this is the cup product. This approach explains the existence of a cup product for cohomology but not for homology: induces a map but would also induce a map , which goes the wrong way round to allow us to define a product.
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