Classifying PL $5$-manifolds up to regular genus seven

Abstract:

In the present paper, we show that the only closed orientable $5$-manifolds of regular genus less or equal than seven are the $5$-sphere ${\mathbb {S}^5}$ and the connected sums of $m$ copies of ${\mathbb {S}^1} \times {\mathbb {S}^4}$, with $m \leqslant 7$. As a consequence, the genus of ${\mathbb {S}^3} \times {\mathbb {S}^2}$ is proved to be eight. This suggests a possible approach to the ($3$-dimensional) Poincaré Conjecture, via the well-known classification of simply connected $5$-manifolds, obtained by Smale and Barden.