added pointer to:

- Antoni Kosinski, chapter IX 6.3 of:
*Differential manifolds*, Academic Press 1993 (pdf, ISBN:978-0-12-421850-5)

Corrected the link for Lorman’s slides.

]]>Changed

homotopy groups $\pm 1 \in \pi_n(S^n)$.

to

homotopy classes $\pm 1 \in \pi_n(S^n)$

]]>added pointer to:

- Michael Hopkins (notes by Akhil Mathew), Lecture 13 in:
*Spectra and stable homotopy theory*, 2012 (pdf, HopkinsMathewStableHomotopyTheory.pdf:file)

added pointer to:

- Gereon Quick,
*The $e$-invariant and the $J$-homomorphism*, lecture notes in:*Advanced algebraic topology*, 2014 (pdf)

added pointer to

- Dustin Clausen,
*$p$-adic J-homomorphisms and a product formula*(arXiv:1110.5851)

where the statement is in Section 2.

]]>I have added pointer to

- Craig Westerland,
*Views on the J-homomorphism*MSRI talk, 2014 (recording, pdf)

But I am looking for a citeable reference on the J-homomorphism as a map of spectra

$ko \overset{\;\;\;\;J\;\;\;\;}{\longrightarrow} Pic(\mathbb{S}) \,.$Westerland’s slides mention this on p. 5. But is there a citeable account that goes with this?

]]>added pointer to

- Arpon Raksit,
*Vector fields and the J-homomorphism*, 2014 (pdf)

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*