# 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)

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].