Products in algebraic topology

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

The cross product

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

The cap product

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

The slant product

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

The cup product

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

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) Template:ISBN, especially chapter 3.