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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 4th 2013

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 4th 2013

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 4th 2013

have copied that paragraph also into the entry stable homotopy groups of spheres (which is badly in need of some genuine content)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 4th 2013
• (edited Nov 4th 2013)

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

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 11th 2013
• (edited Nov 11th 2013)

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.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeNov 12th 2013

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

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeNov 17th 2013
• (edited Nov 17th 2013)

added a few more pointers to discussion of the image of J in terms of $K(1)$/$E(1)$-localization of the sphere spectrum:

Is there anything else?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeSep 30th 2014

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.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMar 21st 2019

• Arpon Raksit, Vector fields and the J-homomorphism, 2014 (pdf)
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 9th 2020
• (edited Oct 9th 2020)

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?

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeOct 9th 2020

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

where the statement is in Section 2.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeNov 28th 2020

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeNov 29th 2020

• CommentRowNumber14.
• CommentAuthorJohn Baez
• CommentTimeDec 10th 2020

Changed

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

to

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

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeDec 12th 2020

Corrected the link for Lorman’s slides.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeDec 14th 2020