Here let me write the real projective space $\mathbb{RP}^n \equiv RP^n$.
I computed that
$$w_1(RP^5)=0,$$
$$w_2(RP^5) \in H^2(RP^5, Z_2) \neq 0,$$ thus it is non-zero.
$$w_3(RP^5)=0,$$
$$w_4(RP^5) \in H^4(RP^5, Z_2) \neq 0,$$ thus it is non-zero.
$$w_k(RP^5) = 0$$ for $k \geq 5.$
(1). What are the 2-dimensional sub-manifold generators (or its Poincaré dual 3-sub-manifold) for $w_2(RP^5) \in H^2(RP^5, Z_2)$?
(2). What are the 4-dimensional sub-manifold generators (or its Poincaré dual 1-sub-manifold) for $w_4(RP^5) \in H^4(RP^5, Z_2)$?
Are the answers, $RP^2$ and $RP^4$ respectively, or something else?