Singular cohomology

Summary

Singular cohomology assigns to a topological space $X$ and an abelian group $G$ the groups $H^n(X; G)$. They are built from functions on singular simplices with values in $G$. It is the dual of singular homology, and it turns continuous maps into homomorphisms in the opposite direction. Its extra structure, the cup product, makes $H^\bullet(X; R)$ a ring, which distinguishes spaces that have the same homology groups.