H-space

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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.

Contents 1 History 2 Examples and Properties 3 See also 4 References

[edit] History

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

[edit] 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].

Adams theorem: S⁰, S¹, S³, S⁷ are the only spheres that are H-spaces.

[edit] See also

• Topological group
• Čech cohomology
• Hopf algebra

[edit] 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