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 S⁰, S¹, S³, S⁷ 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, S⁰, S¹, and S³ are groups (Lie groups) with these multiplications. But S⁷ 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.

See also

• Topological group
• Čech cohomology
• Hopf algebra
• Topological monoid
• H-object

References

• Hatcher, Allen (2002), Algebraic topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0. Section 3.C
• Spanier, Edwin H. (1981), Algebraic topology (Corrected reprint of the 1966 original ed.), New York-Berlin: Springer-Verlag, ISBN 0-387-90646-0
• Stasheff, James Dillon (1963), "Homotopy associativity of H-spaces. I, II", Transactions of the American Mathematical Society, 108: 275–292, 293–312, doi:10.2307/1993609, MR 0158400.
• Stasheff, James (1970), H-spaces from a homotopy point of view, Lecture Notes in Mathematics, vol. 161, Berlin-New York: Springer-Verlag.