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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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up vote 3 down vote accepted

An orientable smooth manifold $X$ admits a $\text{Spin}^c$ structure iff its second Stiefel-Whitney class $w_2 \in H^2(X, \mathbb{F}_2)$ is the reduction of a class $c_1 \in H^2(X, \mathbb{Z})$. This condition is equivalent to the condition that the third integral Stiefel-Whitney class $W_3 = \beta w_2 \in H^3(X, \mathbb{Z})$ vanishes, and I guess this is what you're calling the Dixmier-Douady class of $X$. Here $\beta$ is a Bockstein homomorphism. Note that $W_3$ is always $2$-torsion, and since $H^3(X, \mathbb{Z})$ has no torsion if $\dim X \le 3$, the Dixmier-Douady class is always trivial in these low-dimensional cases.

That leaves the case $\dim X = 4$. Here I don't have a proof. It's clear if $H^3(X, \mathbb{Z})$ has no torsion, which in the compact case by Poincare duality is equivalent to $H_1(X, \mathbb{Z})$ having no torsion. It is always possible to pass to a finite cover with this property, and hence I can at least say that a compact orientable smooth $4$-manifold has a finite cover with a $\text{Spin}^c$ structure. But I don't yet see how to prove the full result.

Edit: A proof for the case $\dim X = 4$ is given in this note by Teichner and Vogt. In the compact case they mention that it is a classical result due to Hirzebruch and Hopf, proven using some Wu class computations.

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