In mathematics, an H-space,[1] 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

he 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.

See also

Topological group

Čech cohomology

Hopf algebra

Topological monoid

Notes

The H in H-space was suggested by Jean-Pierre Serre in recognition of the influence exerted on the subject by Heinz Hopf (see J. R. Hubbuck. "A Short History of H-spaces", History of topology, 1999, pages 747–755).

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, 161, Berlin-New York: Springer-Verlag.

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