Bockstein homomorphim and obstruction of spin-c structure

Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $\beta$ be the $$H^2(\mathbb{Z}_2,M) \to H^3(\mathbb{Z},M),$$ such that $\beta(w_2)$ is the integral cohomology class.

Is this true that for certain dimensions of $M=M^d$, say $d=5$, the existence of such a nontrivial $\beta(w_2)$ indicates the obstruction of the spin$^c$ structure of $M$? How do we show this?

• $SU(3)/SO(3)$ (called the Wu manifold) is a 5-dimensional example of this. – user98602 Jul 19 '18 at 16:17
• Thanks, can you be more explicit? You mean it is an obstruction, only in 5d? – annie heart Jul 19 '18 at 16:19
• An oriented vector bundle $E$ has a $\text{spin}^c$ structure iff $\beta w_2(E) = 0$. You can prove this via obstruction theory, as one possible approach. This is just a fact for all vector bundles over (paracompact) spaces. Now, if $E = TM$, this automatically vanishes for dimensions up to 4 - for easy reasons when the dimension is at most 3. See here for dimension 4. The Wu manifold is a simple 5-manifold that does not carry a spin^c structure, showing these arguments don't generalize. – user98602 Jul 19 '18 at 16:29
• Wu x torus is also not spin^c, so you would get examples in any higher dimension. – user98602 Jul 19 '18 at 16:32
• I believe that is spin when N is even, and is neither spin nor spin-c when N is odd. This seems to be a difficult calculation. If you have some specific reason to care about whether that manifold is spin-c I can write up the calculation as an answer to a separate question. – user98602 Jul 22 '18 at 11:21