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"×"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
and the suspension
of the map from "X"×"Y" to "Z".
The map from "X"*"Y" to "S"("X"×"Y") can be obtained by regarding both sides as a quotient of "X"×"Y"×"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"×"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"×"Y"×"I" to "S"("X"×"Y") factors through "X"*"Y".