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, defined by George W. Whitehead.

The original homomorphism is defined geometrically, and gives a homomorphism

:$J\; colon\; pi\_r\; (mathrm\{SO\}(q))\; o\; pi\_\{r+q\}(S^q)\; ,!$

of abelian groups (for integers "q", and "r" ≥ 2).

The stable "J"-homomorphism in stable homotopy theory gives a homomorphism

:$J\; colon\; pi\_r(mathrm\{SO\})\; o\; pi\_r^S\; ,\; ,!$

where 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.

The image of the "J"-homomorphism was described by harvtxt|Adams|1966 and harvtxt|Quillen|1971 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 mod 8, infinite if "r" is 3 mod 4, and order 1 otherwise. 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 harv|Adams|1966, a homomorphism from the stable homotopy groups to Q/Z. The order of the image is 2 if "r" is 0 or 1 mod 8 and positive (so in this case the "J"-homomorphism is injective). If "r" = 4"n"−1 is 3 mod 4 and positive 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.

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An important step in the theory was the "Adams conjecture" from 1963, which allowed the order of the image of the stable "J"-homomorphism to be determined (it is cyclic; see harvtxt|Switzer|1975|p=488 for details). Frank Adams's conjecture was proved about eight years later by Daniel Quillen. The cokernel of the "J"-homomorphism is of interest for counting exotic spheres.

References

*citation|first=J. F. |last=Adams|title=On the groups J(X) I|journal= Topology |volume=2|year=1963|doi=10.1016/0040-9383(63)90001-6
*citation|first=J. F. |last=Adams|title=On the groups J(X) II|journal= Topology |volume=3|year=1965a|doi=10.1016/0040-9383(65)90040-6
*citation|first=J. F. |last=Adams|title=On the groups J(X) III|journal= Topology |volume=3|year=1965b|doi=10.1016/0040-9383(65)90054-6
*citation|first=J. F. |last=Adams|title=On the groups J(X) IV|journal= Topology |volume=5|year=1966|doi= 10.1016/0040-9383(66)90004-8 citation|title= Correction|journal= Topology |volume=7|year=1968|doi= 10.1016/0040-9383(68)90010-4
*citation|first=D. |last=Quillen|title=The Adams conjecture|journal= Topology |volume=10 |year=1971|pages= 67-80|doi=10.1016/0040-9383(71)90018-8
*citation|first=Robert M. |last=Switzer |title=Algebraic Topology—Homotopy and Homology |publisher=Springer-Verlag |year=1975 |isbn=978-0-387-06758-2
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