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A topological space with multiplication having a two-sided homotopy identity. More precisely, a pointed topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h0460302.png" /> for which a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h0460303.png" /> has been given is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h0460305.png" />-space if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h0460306.png" /> and if the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h0460307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h0460308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h0460309.png" /> are homotopic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603010.png" /> to the identity mapping. The marked point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603011.png" /> is called the homotopy identity of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603013.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603014.png" />. Sometimes the term  "H-space"  is used in a narrower sense: It is required that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603015.png" /> be homotopy associative, i.e. that the mappings
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603016.png" /></td> </tr></table>
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are homotopic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603017.png" />. Sometimes one requires also the existence of a homotopy-inverse. This means that a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603018.png" /> must be given for which the mappings
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A topological space with multiplication having a two-sided homotopy identity. More precisely, a pointed topological space  $  ( X , e) $
 +
for which a continuous mapping  $  m : X \times X \rightarrow X $
 +
has been given is called an  $  H $-
 +
space if  $  m ( e , e ) = e $
 +
and if the mappings  $  X \rightarrow X $,
 +
$  x \mapsto m ( x , e ) $
 +
and  $  x \mapsto m ( e , x ) $
 +
are homotopic $  \mathop{\rm rel} ( e , e ) $
 +
to the identity mapping. The marked point  $  e $
 +
is called the homotopy identity of the  $  H $-
 +
space  $  X $.  
 +
Sometimes the term  "H-space" is used in a narrower sense: It is required that  $  m : X \times X \rightarrow X $
 +
be homotopy associative, i.e. that the mappings
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603019.png" /></td> </tr></table>
+
$$
 +
m \circ ( m \times  \mathop{\rm id} ) , m \circ (  \mathop{\rm id} \times m ) : \
 +
X \times X  \rightarrow  X
 +
$$
  
are homotopic to the constant mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603020.png" />. E.g., for any pointed topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603021.png" /> the [[Loop space|loop space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603022.png" /> is a homotopy-associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603023.png" />-space with homotopy-inverse elements, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603024.png" /> is even a commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603026.png" />-space, i.e. a space for which the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603027.png" />,
+
are homotopic $  \mathop{\rm rel} ( e , e ) $.  
 +
Sometimes one requires also the existence of a homotopy-inverse. This means that a mapping  $  \mu : ( X , e ) \rightarrow ( X , e) $
 +
must be given for which the mappings
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603028.png" /></td> </tr></table>
+
$$
 +
x  \mapsto  m ( x , \mu ( x) ) ,\ \
 +
x  \mapsto  m ( \mu ( x) , x )
 +
$$
  
are homotopic. The cohomology groups of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603029.png" />-space form a [[Hopf algebra|Hopf algebra]].
+
are homotopic to the constant mapping  $  X \mapsto e $.
 +
E.g., for any pointed topological space  $  Y $
 +
the [[Loop space|loop space]]  $  \Omega Y $
 +
is a homotopy-associative  $  H $-
 +
space with homotopy-inverse elements, while  $  \Omega  ^ {2} Y = \Omega ( \Omega Y ) $
 +
is even a commutative  $  H $-
 +
space, i.e. a space for which the mappings  $  X \times X \rightarrow X $,
 +
 
 +
$$
 +
( x , y )  \mapsto  m ( x , y ) ,\ \
 +
( x , y )  \mapsto  m ( y , x )
 +
$$
 +
 
 +
are homotopic. The cohomology groups of an $  H $-
 +
space form a [[Hopf algebra|Hopf algebra]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.M. Boardman,  R.M. Vogt,  "Homotopy invariant algebraic structures on topological spaces" , Springer  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.M. Boardman,  R.M. Vogt,  "Homotopy invariant algebraic structures on topological spaces" , Springer  (1973)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Much of the importance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603030.png" />-spaces (with the axioms of homotopy associativity and of homotopy inverse) comes from the fact that a group structure is induced on the set of homotopy classes of mappings from a space into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046030/h04603031.png" />-space. See [[#References|[a1]]].
+
Much of the importance of $  H $-
 +
spaces (with the axioms of homotopy associativity and of homotopy inverse) comes from the fact that a group structure is induced on the set of homotopy classes of mappings from a space into an $  H $-
 +
space. See [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. Chapt. I, Sect. 6</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. Chapt. I, Sect. 6</TD></TR></table>

Latest revision as of 19:42, 5 June 2020


A topological space with multiplication having a two-sided homotopy identity. More precisely, a pointed topological space $ ( X , e) $ for which a continuous mapping $ m : X \times X \rightarrow X $ has been given is called an $ H $- space if $ m ( e , e ) = e $ and if the mappings $ X \rightarrow X $, $ x \mapsto m ( x , e ) $ and $ x \mapsto m ( e , x ) $ are homotopic $ \mathop{\rm rel} ( e , e ) $ to the identity mapping. The marked point $ e $ is called the homotopy identity of the $ H $- space $ X $. Sometimes the term "H-space" is used in a narrower sense: It is required that $ m : X \times X \rightarrow X $ be homotopy associative, i.e. that the mappings

$$ m \circ ( m \times \mathop{\rm id} ) , m \circ ( \mathop{\rm id} \times m ) : \ X \times X \rightarrow X $$

are homotopic $ \mathop{\rm rel} ( e , e ) $. Sometimes one requires also the existence of a homotopy-inverse. This means that a mapping $ \mu : ( X , e ) \rightarrow ( X , e) $ must be given for which the mappings

$$ x \mapsto m ( x , \mu ( x) ) ,\ \ x \mapsto m ( \mu ( x) , x ) $$

are homotopic to the constant mapping $ X \mapsto e $. E.g., for any pointed topological space $ Y $ the loop space $ \Omega Y $ is a homotopy-associative $ H $- space with homotopy-inverse elements, while $ \Omega ^ {2} Y = \Omega ( \Omega Y ) $ is even a commutative $ H $- space, i.e. a space for which the mappings $ X \times X \rightarrow X $,

$$ ( x , y ) \mapsto m ( x , y ) ,\ \ ( x , y ) \mapsto m ( y , x ) $$

are homotopic. The cohomology groups of an $ H $- space form a Hopf algebra.

References

[1] J.M. Boardman, R.M. Vogt, "Homotopy invariant algebraic structures on topological spaces" , Springer (1973)

Comments

Much of the importance of $ H $- spaces (with the axioms of homotopy associativity and of homotopy inverse) comes from the fact that a group structure is induced on the set of homotopy classes of mappings from a space into an $ H $- space. See [a1].

References

[a1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. I, Sect. 6
How to Cite This Entry:
H-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H-space&oldid=47155
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article