H-Space

An -space, named after Heinz Hopf, and sometimes also called a Hopf space, is a topological space together with a continuous binary operation , such that there exists a point  with the property that the two maps  and  are both homotopic to the identity map  on , through homotopies preserving the point . The element  is called a homotopy identity.

One should note that authors do not always agree on the definition of an -space. In some texts, the maps given by  and  are required to be equal to the identity on . In others, the two maps are required to be homotopic to the identity as above, but the homotopies need not fix the element . Fortunately, we have the comforting fact that for any CW-complex, the three definitions above are equivalent.

For any -space  with homotopy identity , the fundamental group with base-point  is an Abelian group. Taking another base-point  in a path-component of  not containing  may, however, result in a non-Abelian fundamental group.

A deep theorem in homotopy theory known as the Hopf invariant one theorem (sometimes also known as Adams' theorem) states that the only -spheres that are -spaces are , , , and .