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 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 with and with, while for S 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 X×Y×I to S factors through X*Y.