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?