J-homomorphism

Template:Short description 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 Template:Harvs, extending a construction of Template:Harvs.

Definition

Whitehead's original homomorphism is defined geometrically, and gives a homomorphism

J : π_{r} (S O (q)) \to π_{r + q} (S^{q})

of abelian groups for integers q, and $r \geq 2$ . (Hopf defined this for the special case $q = r + 1$ .)

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} \to S^{q - 1}

and the homotopy group $π_{r} (SO (q))$ ) consists of homotopy classes of maps from the r-sphere to SO(q). Thus an element of $π_{r} (SO (q))$ can be represented by a map

S^{r} \times S^{q - 1} \to S^{q - 1}

Applying the Hopf construction to this gives a map

S^{r + q} = S^{r} * S^{q - 1} \to S (S^{q - 1}) = S^{q}

in $π_{r + q} (S^{q})$ , which Whitehead defined as the image of the element of $π_{r} (SO (q))$ 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) \to π_{r}^{S},

where $S O$ 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 Template:Harvs, assuming the Adams conjecture of Template:Harvtxt which was proved by Template:Harvs, as follows. The group $π_{r} (SO)$ 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 Template:Harv. In particular the image of the stable J-homomorphism is cyclic. The stable homotopy groups $π_{r}^{S}$ are the direct sum of the (cyclic) image of the J-homomorphism, and the kernel of the Adams e-invariant Template:Harv, a homomorphism from the stable homotopy groups to $ℚ / ℤ$ . 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 $B_{2 n} / 4 n$ , where $B_{2 n}$ is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because $π_{r} (SO)$ is trivial.

r	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
$π_{r} (SO)$	1	2	1	$ℤ$	1	1	1	$ℤ$	2	2	1	$ℤ$	1	1	1	$ℤ$	2	2
$\| im (J) \|$	1	2	1	24	1	1	1	240	2	2	1	504	1	1	1	480	2	2
$π_{r}^{S}$	$ℤ$	2	2	24	1	1	2	240	2²	2³	6	504	1	3	2²	480×2	2²	2⁴
$B_{2 n}$				¹⁄₆				−¹⁄₃₀				¹⁄₄₂				−¹⁄₃₀

Applications

Template:Harvs 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 $J : π_{n} (S O) \to π_{n}^{S}$ appears in the group Θ_n of h-cobordism classes of oriented homotopy n-spheres (Template:Harvtxt).

References

J-homomorphism

Contents

Definition

Image of the J-homomorphism

Applications

References

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J-homomorphism

Definition

Image of the J-homomorphism

Applications

References

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