Examples for obstructions to a $Spin^c$ structure

Question

For a real vector bundle $E$ of rank 3 on the base manifold $M$, the obstruction to having a $Spin^c$ structure on this vector bundle is given by the integral Stiefel-Whitney classes $W_3 = \beta w_2(E)$, where $\beta$ is the Bockstein homomorphism.

Are there any explicit examples of real vector bundles $E$ of rank 3 on 3-manifolds and on 4-manifolds for which the obstruction class $W_3$ is non-vanishing?

In physics terms, I am asking in what situation an $SO(3)$ gauge field on 3-manifolds or 4-manifolds cannot be lifted to a $U(2) = Spin^c(3)$ gauge field?

Answer 1

If $M$ is an orientable 3-manifold then any real vector bundle $E\rightarrow M$ of rank $\geq 2$ admits a $Spin^c$-structure. This follows since duality shows $H^3(M)$ to be torsion free giving $\beta \omega_2=0$.

Now consider $M=S^1\times \mathbb{R}P^2$. The Kunneth theorem shows that $H^3(S^1\times \mathbb{R}P^2)\cong \mathbb{Z}_2$, generated by an element $\beta (x)$ where $x\in H^2(S^1\times \mathbb{R}P^2;\mathbb{Z}_2)$ and $\beta:H^2(S^1\times \mathbb{R}P^2;\mathbb{Z}_2)\rightarrow H^3(S^1\times \mathbb{R}P^2)$ is the Bockstein connecting map. Moreover obstruction theory show that $x$ classifies some rank 3 real vector bundle $E\rightarrow S^1\times\mathbb{R}P^2$ in that $\omega_2(E)=x$. In particular $\beta(x)\neq 0$ and $E$ does not admit a $Spin^c$-structure.

Similarly if $M$ is an orientable $4$-manifold then it is well known that its tangent bundle admits a $Spin^c$-structure. See for instance the note "All 4-Manifolds Have $Spin^c$-Structures" by Teichner and Vogt (available at http://people.mpim-bonn.mpg.de/teichner/Papers/spin.pdf). It is not true for non-orientable 4-manifolds, however, and the the tangent bundle of of $\mathbb{R}P^2\times\mathbb{R}P^2$, given as an example in the cited note, is easily verified to admit no $Spin^c$-structure.

For an example of an arbitrary rank $3$ real vector bundle over an orientable $4$-manifold that does not admit a $Spin^c$-structure simply note that the vector bundle $E\rightarrow S^1\times \mathbb{R}P^2$ given as my previous example extends to a vector bundle $E'\rightarrow S^1\times \mathbb{R}P^3$ which by the same reasoning admits no $Spin^c$-structure.