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J-homomorphism

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(Redirected from E-invariant) 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. 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.

r 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
π r ( SO ) {\displaystyle \pi _{r}(\operatorname {SO} )} 1 2 1 Z {\displaystyle \mathbb {Z} } 1 1 1 Z {\displaystyle \mathbb {Z} } 2 2 1 Z {\displaystyle \mathbb {Z} } 1 1 1 Z {\displaystyle \mathbb {Z} } 2 2
| im ⁡ ( J ) | {\displaystyle |\operatorname {im} (J)|} 1 2 1 24 1 1 1 240 2 2 1 504 1 1 1 480 2 2
π r S {\displaystyle \pi _{r}^{S}} Z {\displaystyle \mathbb {Z} } 2 2 24 1 1 2 240 2 2 6 504 1 3 2 480×2 2 2
B 2 n {\displaystyle B_{2n}} ⁄6 −⁄30 ⁄42 −⁄30

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

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