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.

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

(2) Wu manifold seems to be a 5-dim manifold with an obstruction of spin-c structure, which is $$ SU(3)/SO(3). $$ Does a more general manifold (for an integer $n$) $$ SU(n)/SO(n) $$ admits spin, or spin-c structures, or do $SU(n)/SO(n)$ have obstructions for them?

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.

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

(2) Wu manifold seems to be a 5-dim manifold with an obstruction of spin-c structure, which is $$ SU(3)/SO(3). $$ Does a more general manifold (for an integer $n$) $$ SU(n)/SO(n) $$ admits spin, or spin-c structures, or do $SU(n)/SO(n)$ have obstructions for them?