- Source: J-homomorphism
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. 488). In particular the image of the stable J-homomorphism is cyclic. The stable homotopy groups
π
r
S
{\displaystyle \pi _{r}^{S}}
are the direct sum of the (cyclic) image of the J-homomorphism, and the kernel of the Adams e-invariant (Adams 1966), a homomorphism from the stable homotopy groups to
Q
/
Z
{\displaystyle \mathbb {Q} /\mathbb {Z} }
. If r is 0 or 1 mod 8 and positive, the order of the image is 2 (so in this case the J-homomorphism is injective). If r is 3 or 7 mod 8, the image is a cyclic group of order equal to the denominator of
B
2
n
/
4
n
{\displaystyle B_{2n}/4n}
, where
B
2
n
{\displaystyle B_{2n}}
is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because
π
r
(
SO
)
{\displaystyle \pi _{r}(\operatorname {SO} )}
is trivial.
Applications
Michael Atiyah (1961) introduced the group J(X) of a space X, which for X a sphere is the image of the J-homomorphism in a suitable dimension.
The cokernel of the J-homomorphism
J
:
π
n
(
S
O
)
→
π
n
S
{\displaystyle J\colon \pi _{n}(\mathrm {SO} )\to \pi _{n}^{S}}
appears in the group Θn of h-cobordism classes of oriented homotopy n-spheres (Kosinski (1992)).
References
Atiyah, Michael Francis (1961), "Thom complexes", Proceedings of the London Mathematical Society, Third Series, 11: 291–310, doi:10.1112/plms/s3-11.1.291, MR 0131880
Adams, J. F. (1963), "On the groups J(X) I", Topology, 2 (3): 181, doi:10.1016/0040-9383(63)90001-6
Adams, J. F. (1965a), "On the groups J(X) II", Topology, 3 (2): 137, doi:10.1016/0040-9383(65)90040-6
Adams, J. F. (1965b), "On the groups J(X) III", Topology, 3 (3): 193, doi:10.1016/0040-9383(65)90054-6
Adams, J. F. (1966), "On the groups J(X) IV", Topology, 5: 21, doi:10.1016/0040-9383(66)90004-8. "Correction", Topology, 7 (3): 331, 1968, doi:10.1016/0040-9383(68)90010-4
Hopf, Heinz (1935), "Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension", Fundamenta Mathematicae, 25: 427–440
Kosinski, Antoni A. (1992), Differential Manifolds, San Diego, CA: Academic Press, pp. 195ff, ISBN 0-12-421850-4
Milnor, John W. (2011), "Differential topology forty-six years later" (PDF), Notices of the American Mathematical Society, 58 (6): 804–809
Quillen, Daniel (1971), "The Adams conjecture", Topology, 10: 67–80, doi:10.1016/0040-9383(71)90018-8, MR 0279804
Switzer, Robert M. (1975), Algebraic Topology—Homotopy and Homology, Springer-Verlag, ISBN 978-0-387-06758-2
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, JSTOR 1968956, MR 0007107
Whitehead, George W. (1978), Elements of homotopy theory, Berlin: Springer, ISBN 0-387-90336-4, MR 0516508