Products in algebraic topology

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

The cross product

[math]\displaystyle{ H_p(X) \otimes H_q(Y) \to H_{p+q}(X\times Y) }[/math]

The cap product

[math]\displaystyle{ \frown\ : H_p(X;R)\times H^q(X;R) \rightarrow H_{p-q}(X;R) }[/math]

The slant product

[math]\displaystyle{ \backslash\ : H_p(X;R)\times H^q(X\times Y;R) \rightarrow H^{q-p}(Y;R) }[/math]

The cup product

[math]\displaystyle{ H^p(X) \otimes H^q(X) \to H^{p+q}(X) }[/math]

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.