Hopf construction

In algebraic topology, the Hopf construction constructs a map from the join [math]\displaystyle{ X*Y }[/math] of two spaces [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] to the suspension [math]\displaystyle{ SZ }[/math] of a space [math]\displaystyle{ Z }[/math] out of a map from [math]\displaystyle{ X\times Y }[/math] to [math]\displaystyle{ Z }[/math]. It was introduced by Hopf (1935) in the case when [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are spheres. (Whitehead 1942) used it to define the J-homomorphism.

Construction

The Hopf construction can be obtained as the composition of a map

[math]\displaystyle{ X*Y\rightarrow S(X\times Y) }[/math]

and the suspension

[math]\displaystyle{ S(X\times Y)\rightarrow SZ }[/math]

of the map from [math]\displaystyle{ X\times Y }[/math] to [math]\displaystyle{ Z }[/math].

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

References

• Hopf, H. (1935), "Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension", Fund. Math. 25: 427–440, http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv25i1p35bwm 
• 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