H-space

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