H-space

A topological space $X$ is said to be an H-space (or Hopf-space) if there exists a 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-abelian for other choices of basepoint, if $X$ is not path-connected.)

Title	H-space
Canonical name	Hspace
Date of creation	2013-03-22 16:18:18
Last modified on	2013-03-22 16:18:18
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	4
Author	yark (2760)
Entry type	Definition
Classification	msc 55P45
Synonym	Hopf-space
Synonym	H space
Synonym	Hopf space