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 , together with an element of and a continuous map , such that and the maps and are both homotopic to the identity map through maps sending to . 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 is an H-space if there exists and such that the triple is an H-space as in the above definition. Alternatively, an H-space may be defined without requiring homotopies to fix the basepoint , or by requiring 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.Hatcher p.287

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'': ,1× ,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, 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
* H-object

Notes

References

*. Section 3.C
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{{DEFAULTSORT:H-Space
Homotopy theory
Algebraic topology
Hopf algebras