J-homomorphism

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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 Whitehead.

The original homomorphism is defined geometrically, and gives a homomorphism

J: π_r(SO(q)) → π_{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: π_r(SO) → π_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.

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 Switzer, Algebraic Topology 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.

[edit] References

• J. F. Adams, On the groups J(X) I, Topology 2, (1963)
• D. Quillen, The Adams conjecture, Topology 10 (1971) 67-80.