- H-space
In
mathematics , an H-space is atopological space "X" (generally assumed to be connected) together with a continuous map μ : "X" × "X" → "X" with anidentity 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 behomotopic 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 areCW complex es. Everytopological group is an H-space; however, in the general case, as compared to a topological group, H-spaces may lackassociativity and inverses.History
The name "H-space" was suggested by
Jean-Pierre Serre in honor ofHeinz Hopf Citation needed Examples and Properties
The multiplicative structure of a H-space adds structure to its homology and
cohomology group s. For example, thecohomology ring of apath-connected H-space with finitely generated and free cohomology groups is aHopf algebra . Also, one can define thePontryagin 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
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