In the present paper we show that the only closed orientable PL 5-manifolds of regular genus less or equal to seven are the 5-sphere $S^5$ and the connected sum of m copies of $S^1 X S^4$, with $m \le 7$. As a consequence, the genus of $S^3 X S^2$ is proved to be eight. This suggests a possible approach to the (3-dimensional) Poincarè Conjecture, via the well-known classification of simply connected 5-manifolds, obtained by Smale and Barden.
Classifying PL 5-manifolds up to regular genus seven / Casali, Maria Rita; Gagliardi, Carlo. - In: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9939. - STAMPA. - 120:(1994), pp. 275-283. [10.1090/S0002-9939-1994-1205484-4]
Classifying PL 5-manifolds up to regular genus seven
CASALI, Maria Rita;GAGLIARDI, Carlo
1994
Abstract
In the present paper we show that the only closed orientable PL 5-manifolds of regular genus less or equal to seven are the 5-sphere $S^5$ and the connected sum of m copies of $S^1 X S^4$, with $m \le 7$. As a consequence, the genus of $S^3 X S^2$ is proved to be eight. This suggests a possible approach to the (3-dimensional) Poincarè Conjecture, via the well-known classification of simply connected 5-manifolds, obtained by Smale and Barden.
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