The present paper adopts a computational approach to the study of nonorientable 3-manifolds: in fact, we describe how to create an automatic catalogue of all nonorientable 3-manifolds admitting coloured triangulations with a fixed number of tetrahedra. In particular, the catalogue has been effectively produced and analysed for up to 26 tetrahedra, to reach the complete classification of all involved 3-manifolds. As a consequence, the following summarising result can be stated: THEOREM I. Exactly seven closed connected prime nonorientable 3-manifolds exist, which admit a coloured triangulation consisting of at most 26 tetrahedra. More precisely, they are the four Euclidean nonorientable 3-manifolds, the nontrivial $S^2$-bundle over $S^1$, the topological product between the real projective plane $RP^2$ and $S^1$, and the torus bundle over $S^1$, with monodromy induced by matrix (0,1;1,-1).
Classification of nonorientable 3-manifolds admitting decompositions into <= 26 coloured tetrahedra / Casali, Maria Rita. - In: ACTA APPLICANDAE MATHEMATICAE. - ISSN 0167-8019. - STAMPA. - 54:(1998), pp. 75-97. [10.1023/A:1006014705085]
Classification of nonorientable 3-manifolds admitting decompositions into <= 26 coloured tetrahedra
CASALI, Maria Rita
1998
Abstract
The present paper adopts a computational approach to the study of nonorientable 3-manifolds: in fact, we describe how to create an automatic catalogue of all nonorientable 3-manifolds admitting coloured triangulations with a fixed number of tetrahedra. In particular, the catalogue has been effectively produced and analysed for up to 26 tetrahedra, to reach the complete classification of all involved 3-manifolds. As a consequence, the following summarising result can be stated: THEOREM I. Exactly seven closed connected prime nonorientable 3-manifolds exist, which admit a coloured triangulation consisting of at most 26 tetrahedra. More precisely, they are the four Euclidean nonorientable 3-manifolds, the nontrivial $S^2$-bundle over $S^1$, the topological product between the real projective plane $RP^2$ and $S^1$, and the torus bundle over $S^1$, with monodromy induced by matrix (0,1;1,-1).Pubblicazioni consigliate
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