H-space

In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed.

Definition

An H-space consists of a topological space X, together with an element e of X and a continuous map μ : X × X → X, such that μ(e, e) = e and the maps x ↦ μ(x, e) and x ↦ μ(e, x) are both homotopic to the identity map through maps sending e to e. This may be thought of as a pointed topological space together with a continuous multiplication for which the basepoint is an identity element up to basepoint-preserving homotopy.

One says that a topological space X is an H-space if there exists e and μ such that the triple (X, e, μ) is an H-space as in the above definition. Alternatively, an H-space may be defined without requiring homotopies to fix the basepoint e, or by requiring e to be an exact identity, without any consideration of homotopy. In the case of a CW complex, all three of these definitions are in fact equivalent.

Examples and properties

The standard definition of the fundamental group, together with the fact that it is a group, can be rephrased as saying that the loop space of a pointed topological space has the structure of an H-group, as equipped with the standard operations of concatenation and inversion. Furthermore a continuous basepoint preserving map of pointed topological space induces a H-homomorphism of the corresponding loop spaces; this reflects the group homomorphism on fundamental groups induced by a continuous map.

It is straightforward to verify that, given a pointed homotopy equivalence from a H-space to a pointed topological space, there is a natural H-space structure on the latter space. As such, the existence of an H-space structure on a given space is only dependent on pointed homotopy type.

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. http://www.math.cornell.edu/~hatcher/AT/ATpage.html. . 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. 
• "Homotopy associativity of H-spaces. I, II", Transactions of the American Mathematical Society 108 (2): 275–292, 293–312, 1963, doi:10.2307/1993609 .
• Stasheff, James (1970), H-spaces from a homotopy point of view, Lecture Notes in Mathematics, 161, Berlin-New York: Springer-Verlag .
• Switzer, Robert M. (1975). Algebraic topology—homotopy and homology.. Die Grundlehren der mathematischen Wissenschaften. 212. New York-Heidelberg: Springer-Verlag.