Classifying PL 5-manifolds up to regular genus seven

Abstract

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