In algebraic topology, several types of products are defined on homological and cohomological theories.
The cross product
H p ( X ) ⊗ H q ( Y ) → H p + q ( X × Y ) {\displaystyle H_{p}(X)\otimes H_{q}(Y)\to H_{p+q}(X\times Y)}
When X and Y are CW complexes, then the product X × Y {\displaystyle X\times Y} has a natural CW structure, and the cross product can be understood as induced by the chain map sending a p-cell e p {\displaystyle e_{p}} in X and a q-cell e q {\displaystyle e_{q}} in Y to the product cell e p × e q {\displaystyle e_{p}\times e_{q}} in X × Y {\displaystyle X\times Y} . An equivalent but slightly more complicated definition can be given for singular homology. The cross product is used to prove the Künneth theorem relating the homology of X and Y to the homology of X × Y {\displaystyle X\times Y} .
The cap product
⌢ : H p ( X ; R ) × H q ( X ; R ) → H p − q ( X ; R ) {\displaystyle \frown \ :H_{p}(X;R)\times H^{q}(X;R)\rightarrow H_{p-q}(X;R)}
The slant product
/ : H p ( X ; R ) × H q ( X × Y ; R ) → H q − p ( Y ; R ) {\displaystyle /:H_{p}(X;R)\times H^{q}(X\times Y;R)\rightarrow H^{q-p}(Y;R)}
The cup product
H p ( X ) ⊗ H q ( X ) → H p + q ( X ) {\displaystyle H^{p}(X)\otimes H^{q}(X)\to H^{p+q}(X)}
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 Singular homology Differential graded algebra: the algebraic structure arising on the cochain level for the cup product Poincaré duality: swaps some of these Intersection theory: for a similar theory in algebraic geometry
References Hatcher, A., Algebraic Topology, Cambridge University Press (2002) ISBN 0-521-79540-0, especially chapter 3.
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