H-space

In mathematics, an H-space, or a topological unital magma, is a topological space "X" (generally assumed to be connected) together with a continuous map μ : "X" × "X" → "X" with an identity element "e" such 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 (in this case "e" is called homotopy identity), sometimes through basepoint preserving maps. These three definitions are in fact equivalent for H-spaces that are CW complexes. Every topological group is an H-space; however, in the general case, as compared to a topological group, H-spaces may lack associativity and inverses.
Examples and properties.
The multiplicative structure of an 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 an H-space.
The fundamental group of an H-space is abelian. To see this, let "X" be an 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"].
Adams' Hopf invariant one theorem, named after Frank Adams, states that "S"0, "S"1, "S"3, "S"7 are the only spheres that are H-spaces. Each of these spaces forms an H-space by viewing it as the subset of norm-one elements of the reals, complexes, quaternions, and octonions, respectively, and using the multiplication operations from these algebras. In fact, "S"0, "S"1, and "S"3 are groups (Lie groups) with these multiplications. But "S"7 is not a group in this way because octonion multiplication is not associative, nor can it be given any other continuous multiplication for which it is a group.