Hopf construction

In algebraic topology, the Hopf construction constructs a map from the join $$X*Y$$ of two spaces $$X$$ and $$Y$$ to the suspension $$SZ$$ of a space $$Z$$ out of a map from $$X\times Y$$ to $$Z$$. It was introduced by in the case when $$X$$ and $$Y$$ are spheres. used it to define the J-homomorphism.

Construction
The Hopf construction can be obtained as the composition of a map
 * $$X*Y\rightarrow S(X\times Y)$$

and the suspension
 * $$S(X\times Y)\rightarrow SZ$$

of the map from $$X\times Y$$ to $$Z$$.

The map from $$X*Y$$ to $$S(X\times Y)$$ can be obtained by regarding both sides as a quotient of $$X\times Y\times I$$ where $$I$$ is the unit interval. For $$X*Y$$ one identifies $$(x,y,0)$$ with $$(z,y,0)$$ and $$(x,y,1)$$ with $$(x,z,1)$$, while for $$S(X\times Y)$$ one contracts all points of the form $$(x,y,0)$$ to a point and also contracts all points of the form $$(x,y,1)$$ to a point. So the map from $$X\times Y\times I$$ to $$S(X\times Y)$$ factors through $$X*Y$$.