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 ${ℝ}^3$.

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:

   • $S^2\times S^1$

   • $ℝP^2\times S^1$

   • $T\times S^1$

   • $K\times S^1$

   where $S^1$ and $S^2$ are the 1- and 2-dimensional spheres respectively, $T$ is a torus, $K$ a Klein bottle, and $ℝP^2$ 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 $[\varphi ]$ 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 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.

Title	3-manifold
Canonical name	3manifold
Date of creation	2013-03-22 15:40:55
Last modified on	2013-03-22 15:40:55
Owner	juanman (12619)
Last modified by	juanman (12619)
Numerical id	17
Author	juanman (12619)
Entry type	Definition
Classification	msc 57N10
Related topic	manifold
Related topic	DehnsLemma
Related topic	SphereTheorem
Related topic	LoopTheorem
Related topic	SeifertFiberSpace
Related topic	Manifold