H-space

In mathematics, an H-space is a topological space X (generally assumed to be connected) together with a continuous map μ : X × X → X with an identity element e so that μ(e, x) = μ(x, e) = x for all x in X. Alternatively, the maps μ(e, x) and μ(x, e) are sometimes only required to be homotopic to the identity, sometimes through basepoint preserving maps. These three definitions are in fact equivalent for H-spaces that are CW complexes. Compared to topological groups, H spaces may lack associativity and inverses.

The name H-space was suggested by Jean-Pierre Serre in honor of Heinz Hopf.

The multiplicative structure of a H-space adds structure to its homology and cohomology groups. For example, the cohomology ring of a path connected H-space with finitely generated and free cohomology groups is a Hopf algebra. Also, one can define the Pontryagin product on the homology groups of a H-space.

The fundamental group of a H-space is abelian. To see this, let X be a H-space with identity e and let f and g be loops at e. Define a map F: [0,1]×[0,1] → X by F(a,b) = f(a)g(b). Then F(a,0) = F(a,1) = f(a)e is homotopic to f, and F(0,b) = F(1,b) = eg(b)is homotopic to g. It is clear how to define a homotopy from [f][g] to [g][f].

The only spheres that are H-spaces are S⁰, S¹, S³, and S⁷.