H-space

A topological space $X$ is said to be an H-space (or Hopf-space)if there existsa continuous binary operation $\phi :X\times X\to X$and a point $p\in X$ such that the functions$X\to X$ defined by $x\mapsto \phi (p,x)$ and $x\mapsto \phi (x,p)$are both homotopic to the identity map via homotopies that leave $p$ fixed.The element $p$ is sometimes referred to as an ‘identity’,although it need not be an identity element in the usual sense.Note that the definition implies that $\phi (p,p)=p$.

Topological groups are examples of H-spaces.

If $X$ is an H-space with ‘identity’ $p$,then the fundamental group ${\pi }_1(X,p)$ is abelian.(However, it is possible for the fundamental group to be non-abelianfor other choices of basepoint, if $X$ is not path-connected.)