Products in algebraic topology

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In algebraic topology, several types of products are defined on homological and cohomological theories.

${\displaystyle H_p(X)\otimes H_q(Y)\to H_{p+q}(X\times Y)}$

When X and Y are CW complexes, then the product ${\displaystyle X\times Y}$ has a natural CW structure, and the cross product can be understood as induced by the chain map sending a p-cell ${\displaystyle e_p}$ in X and a q-cell ${\displaystyle e_q}$ in Y to the product cell ${\displaystyle e_p\times e_q}$ in ${\displaystyle X\times Y}$. 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 ${\displaystyle X\times Y}$.

${\displaystyle ⌢:H_p(X;R)\times H^q(X;R)\to H_{p-q}(X;R)}$

${\displaystyle /:H_p(X;R)\times H^q(X\times Y;R)\to H^{q-p}(Y;R)}$

${\displaystyle H^p(X)\otimes H^q(X)\to H^{p+q}(X)}$

This product can be understood as induced by the exterior product of differential forms in de Rham cohomology.^[1] It makes the singular cohomology of a connected manifold into a unitary supercommutative ring.

• 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

• Hatcher, A., Algebraic Topology, Cambridge University Press (2002) ISBN 0-521-79540-0, especially chapter 3.

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