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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 4th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the plain traditional definition to _[[J-homomorphism]]_

   added the plain traditional definition to J-homomorphism

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 4th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave added the statement of the Adams cojecture to _[[J-homomorphism]]_.

   have added the statement of the Adams cojecture to J-homomorphism.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 4th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave copied that paragraph also into the entry _[[stable homotopy groups of spheres]]_ (which is badly in need of some genuine content)

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 4th 2013
   • (edited Nov 4th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave added the list of values of $\vert J(\pi_{4k-1}(O))\vert $ for low $k$.

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

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 11th 2013
   • (edited Nov 11th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added some more comments to _[J-homomorphism -- Definition -- On groups](http://ncatlab.org/nlab/show/J-homomorphism#OnGroups)_ 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 12th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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](http://ncatlab.org/nlab/show/J-homomorphism#ImageOfJInChromotopy)_

   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

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 17th 2013
   • (edited Nov 17th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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](http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks)_ ([pdf](http://math.mit.edu/conferences/talbot/2013/1-Behrens-intro.pdf), [pdf](http://math.mit.edu/conferences/talbot/2013/Behrens-Introduction-chromatic-homotopy-thy-4-22-13.pdf)) {#Behrens13} * Ben Knudsen, _First chromatic layer of the sphere spectrum = homotopy of the $K(1)$-local sphere_, talk at _[2013 Pre-Talbot Seminar](http://math.northwestern.edu/~htanaka/pretalbot2013/index.php)_ ([pdf](http://math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-04-04-Rob-Legg-K1-local-sphere.pdf)) {#Knudsen13} * [[Vitaly Lorman]], _Chromatic homotopy theory at height 1 and the image of $J$_, talk at _[Talbot 2013: Chromatic Homotopy Theory](http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks)_ ([pdf](http://math.mit.edu/conferences/talbot/2013/Image%20of%20J-1.pdf)) {#Lorman13} Is there anything else?

   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?

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeSep 30th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to [Gaudens-Menichi 07](http://ncatlab.org/nlab/show/J-homomorphism#GaudensMenichi07) 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 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.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMar 21st 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * Arpon Raksit, _Vector fields and the J-homomorphism_, 2014 ([pdf](http://stanford.edu/~arpon/math/files/vfields.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/J-homomorphism/38">diff</a>, <a href="https://ncatlab.org/nlab/revision/J-homomorphism/38">v38</a>, <a href="https://ncatlab.org/nlab/show/J-homomorphism">current</a>

   added pointer to

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

   diff, v38, current

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2020
    • (edited Oct 9th 2020)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have added pointer to * [[Craig Westerland]], _Views on the J-homomorphism_ MSRI talk, 2014 ([recording](https://mathinstitutes.org/videos/videos/1464), [pdf](https://docs.google.com/viewer?url=https://www.msri.org/workshops/685/schedules/17895/documents/1978/assets/20102)) 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? <a href="https://ncatlab.org/nlab/revision/diff/J-homomorphism/41">diff</a>, <a href="https://ncatlab.org/nlab/revision/J-homomorphism/41">v41</a>, <a href="https://ncatlab.org/nlab/show/J-homomorphism">current</a>

    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?

    diff, v41, current

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to * Dustin Clausen, _$p$-adic J-homomorphisms and a product formula_ ([arXiv:1110.5851](https://arxiv.org/abs/1110.5851)) where the statement is in Section 2. <a href="https://ncatlab.org/nlab/revision/diff/J-homomorphism/41">diff</a>, <a href="https://ncatlab.org/nlab/revision/J-homomorphism/41">v41</a>, <a href="https://ncatlab.org/nlab/show/J-homomorphism">current</a>

    added pointer to

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

    where the statement is in Section 2.

    diff, v41, current

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * [[Gereon Quick]], _The $e$-invariant and the $J$-homomorphism_, lecture notes in: _[Advanced algebraic topology](https://folk.ntnu.no/gereonq/Math231br.html)_, 2014 ([pdf](https://folk.ntnu.no/gereonq/Math231b/231b-Lecture30.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/J-homomorphism/43">diff</a>, <a href="https://ncatlab.org/nlab/revision/J-homomorphism/43">v43</a>, <a href="https://ncatlab.org/nlab/show/J-homomorphism">current</a>

    added pointer to:

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

    diff, v43, current

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 13 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]]) <a href="https://ncatlab.org/nlab/revision/diff/J-homomorphism/44">diff</a>, <a href="https://ncatlab.org/nlab/revision/J-homomorphism/44">v44</a>, <a href="https://ncatlab.org/nlab/show/J-homomorphism">current</a>

    added pointer to:

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

    diff, v44, current

14. • CommentRowNumber14.
    • CommentAuthorJohn Baez
    • CommentTimeDec 10th 2020
    • PermaLink
    Author: John Baez
    Format: MarkdownItexChanged [[homotopy groups]] $\pm 1 \in \pi_n(S^n)$. to [[homotopy classes]] $\pm 1 \in \pi_n(S^n)$ <a href="https://ncatlab.org/nlab/revision/diff/J-homomorphism/45">diff</a>, <a href="https://ncatlab.org/nlab/revision/J-homomorphism/45">v45</a>, <a href="https://ncatlab.org/nlab/show/J-homomorphism">current</a>

    Changed

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

    to

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

    diff, v45, current

15. • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 12th 2020
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexCorrected the link for Lorman's slides. <a href="https://ncatlab.org/nlab/revision/diff/J-homomorphism/46">diff</a>, <a href="https://ncatlab.org/nlab/revision/J-homomorphism/46">v46</a>, <a href="https://ncatlab.org/nlab/show/J-homomorphism">current</a>

    Corrected the link for Lorman’s slides.

    diff, v46, current

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeDec 14th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * [[Antoni Kosinski]], chapter IX 6.3 of: _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf), [ISBN:978-0-12-421850-5](https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C)) <a href="https://ncatlab.org/nlab/revision/diff/J-homomorphism/47">diff</a>, <a href="https://ncatlab.org/nlab/revision/J-homomorphism/47">v47</a>, <a href="https://ncatlab.org/nlab/show/J-homomorphism">current</a>

    added pointer to:

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

    diff, v47, current