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

Main article: Cap product

$\frown \ :H_{p}(X;R)\times H^{q}(X;R)\rightarrow H_{p-q}(X;R)$

The slant product

Main article: slant product

$\backslash \ :H_{p}(X;R)\times H^{q}(X\times Y;R)\rightarrow H^{q-p}(Y;R)$

The cup product

Main article: 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) ISBN 0-521-79540-0, especially chapter 3.