In algebraic topology, the Hopf construction constructs a map from the join X ∗ Y {\displaystyle X*Y} of two spaces X {\displaystyle X} and Y {\displaystyle Y} to the suspension S Z {\displaystyle SZ} of a space Z {\displaystyle Z} out of a map from X × Y {\displaystyle X\times Y} to Z {\displaystyle Z} . It was introduced by Hopf (1935) in the case when X {\displaystyle X} and Y {\displaystyle Y} are spheres. Whitehead (1942) used it to define the J-homomorphism.
Construction The Hopf construction can be obtained as the composition of a map
X ∗ Y → S ( X × Y ) {\displaystyle X*Y\rightarrow S(X\times Y)}
and the suspension
S ( X × Y ) → S Z {\displaystyle S(X\times Y)\rightarrow SZ}
of the map from X × Y {\displaystyle X\times Y} to Z {\displaystyle Z} . The map from X ∗ Y {\displaystyle X*Y} to S ( X × Y ) {\displaystyle S(X\times Y)} can be obtained by regarding both sides as a quotient of X × Y × I {\displaystyle X\times Y\times I} where I {\displaystyle I} is the unit interval. For X ∗ Y {\displaystyle X*Y} one identifies ( x , y , 0 ) {\displaystyle (x,y,0)} with ( z , y , 0 ) {\displaystyle (z,y,0)} and ( x , y , 1 ) {\displaystyle (x,y,1)} with ( x , z , 1 ) {\displaystyle (x,z,1)} , while for S ( X × Y ) {\displaystyle S(X\times Y)} one contracts all points of the form ( x , y , 0 ) {\displaystyle (x,y,0)} to a point and also contracts all points of the form ( x , y , 1 ) {\displaystyle (x,y,1)} to a point. So the map from X × Y × I {\displaystyle X\times Y\times I} to S ( X × Y ) {\displaystyle S(X\times Y)} factors through X ∗ Y {\displaystyle X*Y} .
References Hopf, H. (1935), "Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension", Fund. Math., 25: 427–440 Whitehead, George W. (1942), "On the homotopy groups of spheres and rotation groups", Annals of Mathematics, Second Series, 43 (4): 634–640, doi:10.2307/1968956, ISSN 0003-486X, JSTOR 1968956, MR 0007107
