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 Hopf (1935) in the case when and are spheres. Whitehead (1942) used it to define the J-homomorphism.

Construction

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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  .

References

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  • 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