Difference between revisions of "H-space"
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| − | are homotopic | + | 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 | + | 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. The cohomology groups of an | + | 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 | + | 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=18666
H-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H-space&oldid=18666
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article