mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an H-space is a homotopy-theoretic version of a generalization of the notion of

topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...

, in which the axioms on

associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...

and inverses are removed.

Definition

An H-space consists of a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

, together with an element of and a

continuous map In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

, such that and the maps and are both homotopic to the

identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

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 In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

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 In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...

, all three of these definitions are in fact equivalent.

Examples and properties

The standard definition of the

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

, 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 In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...

on fundamental groups induced by a continuous map. It is straightforward to verify that, given a pointed

homotopy equivalence In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...

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 group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

s. For example, the

cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually un ...

of a

path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...

H-space with finitely generated and free cohomology groups is a

Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...

. Also, one can define the Pontryagin product on the homology groups of an H-space.Hatcher p.287 The

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 The Synchronized Position Hold Engage and Reorient Experimental Satellite (SPHERES) are a series of miniaturized satellites developed by MIT's Space Systems Laboratory for NASA and US Military, to be used as a low-risk, extensible test bed for t ...

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,

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

s, and

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...

s, respectively, and using the multiplication operations from these algebras. In fact, ''S''⁰, ''S''¹, and ''S''³ are groups (

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

s) 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.

Notes

References

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

Definition

Examples and properties

See also

Notes

References