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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Authors: Maria Rita Casali and Carlo Gagliardi
Journal: Proc. Amer. Math. Soc. 120 (1994), 275-283
MSC: Primary 57Q99
MathSciNet review: 1205484
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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.

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Keywords: $ 5$-Manifolds, regular genus, classification, products of spheres and manifolds, handlebodies, Poincaré Conjecture, edge-coloured graphs, $ 2$-cell embeddings
Article copyright: © Copyright 1994 American Mathematical Society