Tags: #homotopy/stable-homotopy
J-homomorphism
References
What is the \(J\) homomorphism?
The \(J\) homomorphism seems to link the Homotopy Groups of Spheres. For example \(J\) takes \(\pi_k(SL^n(R))\) to \(\pi_{n}^k S^n\).
Look at the map \begin{align*} SO(n) &\to \Omega^n S^n\\ A: ({\mathbf{R}}^n \to {\mathbf{R}}^n) &\mapsto A^+ \end{align*}
AKA \begin{align*} J: \pi_n(SO(k)) \to \pi_{n+k}(S_k) \end{align*}
Where we view a matrix as a linear function on \({\mathbf{R}}^n\), and take it to its compactification which is a map \(S^n\to S^n\). Taking the limit yields a map from \({\operatorname{SO}}^\infty \to QS^0 = \Omega^\infty \Sigma^\infty S^0\), and taking \(\pi_0\) of both sides induces the \(J\) homomorphism. The RHS is equal to \(\pi_*^s\), the stable homotopy groups of spheres. But the homotopy groups of \(SO\) were computed by Bott, and have some 8-fold periodicity.
The image of \(J\) was found by Adams in ’66 or so, it is a finite group with order the denominator of some function involving Bernoulli numbers. However, the pattern is more apparent by looking at the \(p\)-stems, then the number of connected dots really just depends on the \(p\)-adic divisibility of the horizontal number plus 1. The image of \(J\) is just the bottom row in these stem diagrams.
