In algebraic topology, several types of products are defined on homological and cohomological theories.
The cross product
When X and Y are CW complexes, then the product has a natural CW structure, and the cross product can be understood as induced by the chain map sending a p-cell in X and a q-cell in Y to the product cell in . 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 .
The cap product
The slant product
The cup product
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) , especially chapter 3.
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