Editing H-space

In [[mathematics]], an '''H-space'''<ref>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).</ref> is a [[homotopy theory|homotopy-theoretic]] version of a generalization of the notion of [[topological group]], in which the axioms on [[associativity]] and [[inverse element|inverses]] are removed.

==Definition==
An H-space consists of a [[topological space]] {{mvar|X}}, together with an element {{mvar|e}} of {{mvar|X}} and a [[continuous function (topology)|continuous map]] {{math|μ : ''X'' × ''X'' → ''X''}}, such that {{math|μ(''e'', ''e'') {{=}} ''e''}} and the maps {{math|''x'' ↦ μ(''x'', ''e'')}} and {{math|''x'' ↦ μ(''e'', ''x'')}} are both [[homotopic]] to the [[Identity function|identity map]] through maps sending {{mvar|e}} to {{mvar|e}}.<ref>Spanier p.34; Switzer p.14</ref> 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 {{mvar|X}} is an H-space if there exists {{mvar|e}} and {{math|μ}} such that the triple {{math|(''X'', ''e'', μ)}} is an H-space as in the above definition.<ref>Hatcher p.281</ref> Alternatively, an H-space may be defined without requiring homotopies to fix the basepoint {{mvar|e}}, or by requiring {{mvar|e}} to be an exact identity, without any consideration of homotopy.<ref>Stasheff (1970), p.1</ref> In the case of a [[CW complex]], all three of these definitions are in fact equivalent.<ref>Hatcher p.291</ref>

==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.<ref>Spanier pp.37-39</ref> 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.<ref>Spanier pp.37-39</ref>

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.<ref>Spanier pp.35-36</ref> 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 group|homology]] and [[cohomology group]]s.  For example, the [[cohomology ring]] of a [[path-connected]] H-space with finitely generated and free cohomology groups is a [[Hopf algebra]].<ref>Hatcher p.283</ref>  Also, one can define the [[Pontryagin product]] on the homology groups of an H-space.<ref>Hatcher p.287</ref>

The [[fundamental group]] of an H-space is [[abelian group|abelian]].  To see this, let ''X'' be an H-space with identity ''e'' and let ''f'' and ''g'' be [[loop (topology)|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|Hopf invariant one]] theorem, named after [[Frank Adams]], states that ''S''<sup>0</sup>, ''S''<sup>1</sup>, ''S''<sup>3</sup>, ''S''<sup>7</sup> are the only [[n-sphere|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 [[real number|reals]], [[complex number|complexes]], [[quaternion]]s, and [[octonion]]s, respectively, and using the multiplication operations from these algebras. In fact, ''S''<sup>0</sup>, ''S''<sup>1</sup>, and ''S''<sup>3</sup> are groups ([[Lie group]]s) with these multiplications. But ''S''<sup>7</sup> 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==
{{reflist}}

==References==
*{{cite book| last=Hatcher |first= Allen |author-link=Allen Hatcher|title=Algebraic topology |url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html |year= 2002 |publisher=Cambridge University Press |place=Cambridge |isbn=0-521-79540-0}}.  Section 3.C
*{{cite book|last=Spanier|first=Edwin H.|title=Algebraic topology|edition=Corrected reprint of the 1966 original|publisher=Springer-Verlag|location=New York-Berlin|year=1981|isbn=0-387-90646-0}}
*{{citation
 | last = Stasheff | first = James Dillon |author-link=Jim Stasheff
 | journal = [[Transactions of the American Mathematical Society]]
 | mr = 0158400
 | pages = 275–292, 293–312
 | title = Homotopy associativity of ''H''-spaces. I, II
 | volume = 108
 | year = 1963
 | issue = 2 | doi=10.2307/1993609| jstor = 1993609 }}.
* {{citation| last=Stasheff|first= James|author-link=Jim Stasheff|title= H-spaces from a homotopy point of view|series=Lecture Notes in Mathematics|volume=161|publisher=Springer-Verlag|place=Berlin-New York| year = 1970}}.
*{{cite book|last=Switzer|first=Robert M.|title=Algebraic topology—homotopy and homology.|series=Die Grundlehren der mathematischen Wissenschaften|volume=212|publisher=Springer-Verlag|location=New York-Heidelberg|year=1975}}

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[[Category:Homotopy theory]]
[[Category:Algebraic topology]]
[[Category:Hopf algebras]]