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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.

Classifying PL 5-manifolds up to regular genus seven