One can see from simple constructions the great variety of objects that indicate that they are worth to study.
First without boundary:
For example, with the cartesian product we can get:
Or interchanging the roles, bundles as:
For the third class above one can use an orbifold instead of a simple surface to get a class of 3-manifolds called Seifert fiber spaces which are a large class of spaces needed to understand the modern classifications for 3-manifolds.
] J.C. Gómez-Larrañaga. 3-manifolds which are unions of three solid tori, Manuscripta Math. 59 (1987), 325-330.
] J.C. Gómez-Larrañaga, F.J. González-Acuña, J. Hoste. Minimal Atlases on 3-manifolds, Math. Proc. Camb. Phil. Soc. 109 (1991), 105-115.
] J. Hempel. 3-manifolds, Princeton University Press 1976.
] P. Orlik. Seifert Manifolds, Lecture Notes in Math. 291, 1972 Springer-Verlag.
] P. Scott. The geometry of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401-487.
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