J-homomorphism
special homomorphism from a homotopy group of a special orthogonal group to a homotopy group of spheres
In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by George W. Whitehead (1942), extending a construction of Heinz Hopf (1935).
Definition
Whitehead's original homomorphism is defined geometrically, and gives a homomorphism
J
:
π
r
(
S
O
(
q
)
)
→
π
r
+
q
(
S
q
)
{\displaystyle J\colon \pi _{r}(\mathrm {SO} (q))\to \pi _{r+q}(S^{q})}
of abelian groups for integers q, and
r
≥
2
{\displaystyle r\geq 2}
. (Hopf defined this for the special case
q
=
r
+
1
{\displaystyle q=r+1}
.)
The J-homomorphism can be defined as follows.
An element of the special orthogonal group SO(q) can be regarded as a map
S
q
−
1
→
S
q
−
1
{\displaystyle S^{q-1}\rightarrow S^{q-1}}
and the homotopy group
π
r
(
SO
(
q
)
)
{\displaystyle \pi _{r}(\operatorname {SO} (q))}
) consists of homotopy classes of maps from the r-sphere to SO(q).
Thus an element of
π
r
(
SO
(
q
)
)
{\displaystyle \pi _{r}(\operatorname {SO} (q))}
can be represented by a map
S
r
×
S
q
−
1
→
S
q
−
1
{\displaystyle S^{r}\times S^{q-1}\rightarrow S^{q-1}}
Applying the Hopf construction to this gives a map
S
r
+
q
=
S
r
∗
S
q
−
1
→
S
(
S
q
−
1
)
=
S
q
{\displaystyle S^{r+q}=S^{r}*S^{q-1}\rightarrow S(S^{q-1})=S^{q}}
in
π
r
+
q
(
S
q
)
{\displaystyle \pi _{r+q}(S^{q})}
, which Whitehead defined as the image of the element of
π
r
(
SO
(
q
)
)
{\displaystyle \pi _{r}(\operatorname {SO} (q))}
under the J-homomorphism.
Taking a limit as q tends to infinity gives the stable J-homomorphism in stable homotopy theory:
J
:
π
r
(
S
O
)
→
π
r
S
,
{\displaystyle J\colon \pi _{r}(\mathrm {SO} )\to \pi _{r}^{S},}
where
S
O
{\displaystyle \mathrm {SO} }
is the infinite special orthogonal group, and the right-hand side is the r-th stable stem of the stable homotopy groups of spheres.
Image of the J-homomorphism
The image of the J-homomorphism was described by Frank Adams (1966), assuming the Adams conjecture of Adams (1963) which was proved by Daniel Quillen (1971), as follows. The group
π
r
(
SO
)
{\displaystyle \pi _{r}(\operatorname {SO} )}
is given by Bott periodicity. It is always cyclic; and if r is positive, it is of order 2 if r is 0 or 1 modulo 8, infinite if r is 3 or 7 modulo 8, and order 1 otherwise (Switzer 1975, p.
Begin with the source’s own compact description: “J-homomorphism” is special homomorphism from a homotopy group of a special orthogonal group to a homotopy group of spheres. The dossier treats that line as a proposition to test through J-homomorphism, special and homomorphism, not as a finished interpretation.
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