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cup product (Definition)

Let $ X$ be a topological space and $ R$ be a commutative ring. The diagonal map $ \Delta: X \to X \times X$ induces a chain map between singular cochain complexes:

$\displaystyle \Delta^*: C^*(X \times X;\, R) \to C^*(X; \, R) $
.

Let $ h:C^*(X ;\, R) \otimes C^*(X ;\, R) \to C^*(X \times X;\, R )$

denote the chain homotopy equivalence associated with the Kunneth Formula.

Given $ \alpha \in C^p (X ;\, R)$ and $ \beta \in C^q(X ;\, R)$ we define

$ \alpha \smile \beta = \Delta^* h(\alpha \otimes \beta) $.

As $ \Delta^*$ and $ h$ are chain maps, $ \smile$ induces a well defined product on cohomology groups, known as the cup product. Hence the direct sum of the cohomology groups of $ X$ has the structure of a ring. This is called the cohomology ring of $ X$.



"cup product" is owned by whm22.
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Keywords:  homology, homological algebra
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Cross-references: cohomology, ring, structure, direct sum, cohomology groups, product, well defined, chain homotopy equivalence, cochain complexes, singular, chain map, induces, diagonal map, commutative ring, topological space
There are 6 references to this entry.

This is version 4 of cup product, born on 2006-01-05, modified 2006-12-04.
Object id is 7554, canonical name is CupProduct.
Accessed 1871 times total.

Classification:
AMS MSC: 55N45 (Algebraic topology :: Homology and cohomology theories :: Products and intersections)
Pending Errata and Addenda
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