H-space (original) (raw)

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In mathematics, an H-space[1] is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed.

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.[2] 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.[3] 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.[4] In the case of a CW complex, all three of these definitions are in fact equivalent.[5]

Examples and properties

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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.[6] 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.[7]

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.[8] 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.[9] Also, one can define the Pontryagin product on the homology groups of an H-space.[10]

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.

  1. ^ 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).
  2. ^ Spanier p.34; Switzer p.14
  3. ^ Hatcher p.281
  4. ^ Stasheff (1970), p.1
  5. ^ Hatcher p.291
  6. ^ Spanier pp.37-39
  7. ^ Spanier pp.37-39
  8. ^ Spanier pp.35-36
  9. ^ Hatcher p.283
  10. ^ Hatcher p.287