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.

]]>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 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*

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.

have added the list of values of $\vert J(\pi_{4k-1}(O))\vert$ for low $k$.

]]>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 statement of the Adams cojecture to *J-homomorphism*.

added the plain traditional definition to *J-homomorphism*