Given a $M=\frac{\mathbb{CP}^2\times S^1}{\tau}$, where $τ$ acts as $−1$ on the sphere $S^1$ and a complex conjugation on complex projective space $\mathbb{CP}^2$. See Dold, Albrecht (1956), "Erzeugende der Thomschen Algebra N", Mathematische Zeitschrift, 65 (1): 25–35, doi:10.1007/BF01473868.
I think that $$H^1(M,\mathbb{Z}_2)=\mathbb{Z}_2.$$ Let us call the generator as $L$.
what is the Poincaré dual (PD) of this generator $L$ for $H^1(M,\mathbb{Z}_2)$, say PD($L$) as a 4-manifold?
Is this PD($L$)=$\mathbb{CP}^2$?