The 
-homomorphism
is a group homomorphism from homotopy
groups of the orthogonal group 
to homotopy groups of
a hypersphere,
 |
where 
and 
are positive integers.
To define it, write 
for the closed unit ball in Euclidean
space 
and represent a class by a continuous map 
taking the basepoint
to the identity element. On 
, the map 
on 
, together with collapse of 
to the basepoint,
defines a map to 
. Its homotopy
class is ![J_n([f])](/images/equations/J-Homomorphism/Inline13.svg)
(Hatcher 2017, p. 108).
These group homomorphisms are compatible with stabilization and give the stable 
-homomorphism
 |
where 
is the union of the orthogonal groups under the
inclusions 
and 
is the stable homotopy group of spheres
in degree 
(Hatcher 2017, p. 109).
-Theory." Version 2.2, Nov 2017.