# 3-manifold

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

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

First without boundary:

1. 1.

For example, with the cartesian product we can get:

• $S^{2}\times S^{1}$

• ${\bf R}P^{2}\times S^{1}$

• $T\times S^{1}$

2. 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. 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 $[\phi]$ 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 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.

• [G

] J.C. Gómez-Larrañaga. 3-manifolds which are unions of three solid tori, Manuscripta Math. 59 (1987), 325-330.

• [GGH

] J.C. Gómez-Larrañaga, F.J. González-Acuña, J. Hoste. Minimal Atlases on 3-manifolds, Math. Proc. Camb. Phil. Soc. 109 (1991), 105-115.

• [H

] J. Hempel. 3-manifolds, Princeton University Press 1976.

• [O

] P. Orlik. Seifert Manifolds, Lecture Notes in Math. 291, 1972 Springer-Verlag.

• [S

] P. Scott. The geometry of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401-487.

Title 3-manifold 3manifold 2013-03-11 19:23:28 2013-03-11 19:23:28 juanman (12619) (0) 1 juanman (0) Definition