3-manifold

3-manifoldA 3 dimensional manifold is a topological space which is locally homeomorphic to the euclidean space ${𝐑}^3$.

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

First without boundary:

1. For example, with the cartesian product we can get:

   • $S^2\times S^1$

   • $𝐑P^2\times S^1$

   • $T\times S^1$

   • …

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:

   $$S^1\subset E\to F$$

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

   From the previuos paragraph we infer that the mapping class group play a important role inthe 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.