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 Hopf (1935) in the case when X and 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)

and the suspension

S(X×Y) → S(Z)

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.

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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

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