H-space

Summary

An H-space is a pointed space with a continuous multiplication for which the basepoint is a two-sided unit up to homotopy. Topological groups and loop spaces are the main examples. The multiplication gives the cohomology of an H-space extra structure, such as a Hopf algebra structure. This strongly restricts which spaces can be H-spaces. By a theorem of Adams, the only spheres that are H-spaces are $S^0$, $S^1$, $S^3$ and $S^7$.