In algebraic topology, the Hopf construction constructs a map from the join of two spaces ' and ' to the suspension ' of a space out of a map from to '. It was introduced by in the case when ' and ' are spheres. used it to define the J-homomorphism.
The Hopf construction can be obtained as the composition of a map
and the suspension
of the map from to '.
The map from to ' can be obtained by regarding both sides as a quotient of where ' is the unit interval. For one identifies with and with , while for ' one contracts all points of the form to a point and also contracts all points of the form to a point. So the map from to ' factors through .