A topological spaceMathworldPlanetmath X is said to be an H-spaceMathworldPlanetmath (or Hopf-space) if there exists a continuous binary operation φ:X×XX and a point pX such that the functions XX defined by xφ(p,x) and xφ(x,p) are both homotopicMathworldPlanetmathPlanetmath to the identity map via homotopiesMathworldPlanetmath that leave p fixed. The element p is sometimes referred to as an ‘identityPlanetmathPlanetmathPlanetmath’, although it need not be an identity elementMathworldPlanetmath in the usual sense. Note that the definition implies that φ(p,p)=p.

Topological groupsMathworldPlanetmath are examples of H-spaces.

If X is an H-space with ‘identity’ p, then the fundamental groupMathworldPlanetmathPlanetmath π1(X,p) is abelianMathworldPlanetmath. (However, it is possible for the fundamental group to be non-abelianMathworldPlanetmathPlanetmath 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