Hopf construction

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

Construction

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

${\displaystyle X*Y\rightarrow S(X\times Y)}$

and the suspension

${\displaystyle S(X\times Y)\rightarrow SZ}$

of the map from ${\displaystyle X\times Y}$ to ${\displaystyle Z}$.

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

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