I remember that I had asked a question about obstructions for a $n^2$ dimensional vector bundle for which the fibers admit a structure of matrix algebra. I do not remember whether it was an independent question or it was included in comments in another question.

I searched a lot both in web and in Mo to find that question but I can not find it. I remember a participant says that there are obstructions as Diximier Duady classes in 3th cohohomology of the base space.

I would like to read that conversation as a supplementary to the following my MO question

Parallel transport as algebra isomorphism

Can one help my to find that conversation?

Thank you very much

  • 2
    $\begingroup$ I have tried searching using this SEDE query for the word Douady in comments on your posts. The only question if found was Differential and pre-differential of Jacobi identity. (I did not find any mentioning Dixmier.) The same query on math.SE returns no results. (You can try other reasonable keywords.) $\endgroup$ Commented May 6, 2017 at 7:27
  • $\begingroup$ @MartinSleziak Thank you very much for your help. $\endgroup$ Commented May 6, 2017 at 7:47
  • $\begingroup$ But I guess that the question I linked to is not the one you are looking for, right? $\endgroup$ Commented May 6, 2017 at 7:48
  • $\begingroup$ @MartinSleziak Yes it is not but I thank you for showing me the explor data search system. My conversation was about obstructions for fiberwise matrix algebra structure for a vector bundle. $\endgroup$ Commented May 6, 2017 at 7:50
  • $\begingroup$ @MartinSleziak I remember that some one answered me Duaddy Diximier invariant (Or class) or some thing like this(In the third cohomology) $\endgroup$ Commented May 6, 2017 at 7:52
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    $\begingroup$ Are you sure it was spelled Diximier in those comments? (You have also used various spellings of Douady in your question and your comments here on meta.) $\endgroup$ Commented May 6, 2017 at 7:55
  • $\begingroup$ @MartinSleziak Yes you are right. My spelling was incorrect. Thanks :) $\endgroup$ Commented May 6, 2017 at 10:02
  • $\begingroup$ I can't help with finding the MO conversation, but indeed such bundles are governed by Dixmier-Douady characteristic classes in $H^3(X,\mathbb{Z})$. This is all discussed in Husemöller, Joachim, Jurco, Schottenloher "Basic Bundle Theory and K-Cohomology Invariants" (2008), more specifically in Chapter IV. $\endgroup$
    – M.G.
    Commented May 11, 2017 at 22:28
  • $\begingroup$ @July I sincerely thank you very much for this reference. $\endgroup$ Commented May 12, 2017 at 5:07


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