Hopf construction explained

X*Y

of two spaces X

and Y

to the suspension SZ

of a space

out of a map from

X x Y

to Z

. It was introduced by in the case when X

and Y

are spheres. used it to define the J-homomorphism.

Construction

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

X*Y → S(X x Y)

and the suspension

S(X x Y) → SZ

of the map from
X x Y

to

The map from

X*Y

to S(X x Y)

can be obtained by regarding both sides as a quotient of

X x Y x 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 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 x Y x I

to S(X x Y)

factors through

X*Y