On the genus of $\mathbb{R}\mathbf{P}^3\times\mathbb{S}^1$

We continue the topological classification of closed connected orientable 4-manifolds according to the (regular) genus, as developed in a series of papers (see [3], [4], [5]). In particular, we prove that any closed prime orientable PL 4-manifold of genus six is topologically homeomorphic to a lens-fiber bundle over the 1-sphere. There are good reasons to conjecture that the genus six characterizes the topological product $\mathbb{R}\mathbf{P}^3\times\mathbb{S}^1$ of the real projective 3-space by the 1-sphere among closed connected prime orientable 4-manifolds.