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As far as I know, Dixmier-Douady classes represent obstrucions to spin$^c$ structures. Questions:

  1. Could somebody prove or give a reference: manifolds of dimension lower than $5$ always have a vanishing Dixmier-Douady class.

  2. I want to compute Dixmier-Douady classes for $n$-manifolds, $n\geq 5$. Rather, I want to be sure that I understand at a computational level, what DD classes are. Could somebody provide an operational definition of Dixmier-Douady classes that would work for this task? I have no idea how to begin with. E.g. I only know the example in T. Friedrich's book. He considers the homogeneous space $X^5=SU(3)/SO(3)$, shows via homotopy theory that the frame bundle $Q\to X^5$ has vanishing fundamental group and therefore admits no spin$^c$ structure. Indirectly, he is showing that the Dixmier-Douady class of $X^5$ is not trivial. But, can one do the same for arbitrary manifods? could somebody give a simple example how to procede?

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