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 (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.

History

The name "H-space" was suggested by Jean-Pierre Serre in honor of Heinz Hopf^{Citation needed}

Examples and Properties

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 an 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"] .

ee also

*Topological group
*Cech-cohomology
*Hopf algebra