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added the plain traditional definition to J-homomorphism
have added the statement of the Adams cojecture to J-homomorphism.
have copied that paragraph also into the entry stable homotopy groups of spheres (which is badly in need of some genuine content)
have added the list of values of $\vert J(\pi_{4k-1}(O))\vert$ for low $k$.
I have added some more comments to J-homomorphism – Definition – On groups meant to be careful about the argument of how the continuous action of the topological group $O(n)$ on the topological space $S^n$ turns into an $\infty$-action of the homotopy type of the stable orthogonal group on the sphere spectrum.
added the characterization of the image of J in terms of chromatic homotopy theory (in the $E(1)$-local sphere spectrum) to Image of J – In terms of chromatic homotopy theory
added a few more pointers to discussion of the image of J in terms of $K(1)$/$E(1)$-localization of the sphere spectrum:
Mark Behrens, section 1 of Introduction talk at Talbot 2013: Chromatic Homotopy Theory (pdf, pdf) {#Behrens13}
Ben Knudsen, First chromatic layer of the sphere spectrum = homotopy of the $K(1)$-local sphere, talk at 2013 Pre-Talbot Seminar (pdf) {#Knudsen13}
Vitaly Lorman, Chromatic homotopy theory at height 1 and the image of $J$, talk at Talbot 2013: Chromatic Homotopy Theory (pdf) {#Lorman13}
Is there anything else?
added pointer to Gaudens-Menichi 07 for expressing the canonical $O(n)$-action on general $n$-fold loop spaces in terms of the J-homomorphisms.
If anyone has more pointers for this, let me know.
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