3-manifold

In this brief note we define and give instances of the notion of a 3-manifold.

A 3-manifold is a Hausdorff topological space which is locally homeomorphic to the Euclidean space ${\mathbb{R}}^3$ .

One can see from simple constructions the great variety of objects that indicate how they are worth to study.

First examples without boundary:

1. For example, with the Cartesian product we can get:
   • $S^2\times S^1$
   • ${\mathbb{R}}P^2\times S^1$
   • $T\times S^1$
   • $K\times S^1$
   where $S^1$ and $S^2$ are the 1- and 2-dimensional spheres respectively, $T$ is a torus, $K$ a Klein bottle, and $\mathbb{R}P^2$ is the 2-dimensional real projective space.
2. Also by the generalization of the Cartesian product: fiber bundles, one can build bundles $E$ of the type $$F\subset E\to S^1$$ where $F$ is any closed surface.
3. Or interchanging the roles, bundles as: $$S^1\subset E\to F$$
4. knots and links complements

For the second type it is known that for each isotopy class $[\phi]$ of maps $F\to F$ correspond to an unique bundle $E_{\phi}$ . Any homeomorphism $f:F\to F$ representing the isotopy class $[\phi]$ is called a monodromy for $E_{\phi}$ .

From the previous paragraph we infer that the mapping class group play a important role in the understanding at least for this subclass of objets.

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.

References

• 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.