We know [$H^d(\mathbb{RP}^5,\mathbb{Z}_2)=\mathbb{Z}_2$](https://topospaces.subwiki.org/wiki/Cohomology_of_real_projective_space#Coefficients_in_a_module_over_a_2-divisible_ring). So there are two classes of $\mathbb{Z}_2$ generators, trivial and nontrivial, for $d=0,1,2,3,4,5$.


Wha are the Poincaré dual $(5-d)$-dimensional manifolds of the generators PD($a^d$) of $a^d\in H^d(\mathbb{RP}^5,\mathbb{Z}_2)$, for $d=0,1,2,3,4,5$?

\begin{array}{|c|c|}
\hline
\text{PD}(a^d)& a^d =0 \in H^d(\mathbb{RP}^5,\mathbb{Z}_2) &a^d \neq 0 \in H^d(\mathbb{RP}^5,\mathbb{Z}_2)\\ \hline
d=0  &?& \mathbb{RP}^5?\\ 
\hline
d=1 &? &\mathbb{RP}^4?\\ 
\hline
d=2  &?& \mathbb{RP}^3?\\ 
\hline
d=3  &?&\mathbb{RP}^2?\\ 
\hline
d=4  &?&\mathbb{RP}^1?\\ 
\hline
d=5  &?&\mathbb{RP}^0=\text{a point}?\\
\hline
\end{array}

- It is said that if $a^d=0$. A Poincaré dual for $a\in H^d(M;\mathbb{Z}/2)$ is any embedded closed $(5-d)$-manifold which bounds, such as a small $S^{5-d}$ around a point, and this is orientable. Is this a correct generator for the trivial class $a^d =0 \in H^d(\mathbb{RP}^5,\mathbb{Z}_2)$?