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| | In [[algebraic topology]], several types of products are defined on homological and cohomological theories. |
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| | ==The cross product== |
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| | <math>H_p(X) \otimes H_q(Y) \to H_{p+q}(X\times Y)</math> |
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| | ==The cap product== |
| | {{Main| Cap product}} |
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| | :<math>\frown\ : H_p(X;R)\times H^q(X;R) \rightarrow H_{p-q}(X;R)</math> |
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| | ==The slant product== |
| | {{Main| slant product}} |
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| | :<math>\frown\ : H_p(X;R)\times H^q(X\times Y;R) \rightarrow H^{q-p}(Y;R)</math> |
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| | ==The cup product== |
| | {{Main| cup product}} |
| | |
| | :<math>H^p(X) \otimes H^q(X) \to H^{p+q}(X)</math> |
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| | This product can be understood as induced by the exterior product of [[differential form]]s in [[de Rham cohomology]]. It makes the singular cohomology of a connected manifold into a unitary supercommutative ring. |
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| | ==See also== |
| | *[[Singular homology]] |
| | *[[Differential graded algebra]]: the algebraic structure arising on the cochain level for the cup product |
| | *[[Poincaré duality]]: swaps some of these |
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| | *[[Intersection theory]]: for a similar theory in [[algebraic geometry]] |
| | |
| | [[Category:Algebraic topology]] |
| | [[Category:Homology theory]] |
| | [[Category:Binary operations]] |
Revision as of 23:47, 11 December 2013
Template:Orphan
In algebraic topology, several types of products are defined on homological and cohomological theories.
The cross product
The cap product
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The slant product
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
The cup product
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
This product can be understood as induced by the exterior product of differential forms in de Rham cohomology. It makes the singular cohomology of a connected manifold into a unitary supercommutative ring.
See also