3manifold
In this brief note we define and give instances of the notion of a 3manifold.
A 3manifold 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 2dimensional spheres respectively, $T$ is a torus, $K$ a Klein bottle, and $\mathbb{R}{P}^{2}$ is the 2dimensional 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 3manifolds called Seifert fiber spaces which are a large class of spaces needed to understand the modern classifications for 3manifolds.

•
J.C. GómezLarrañaga. 3manifolds which are unions of three solid tori, Manuscripta Math. 59 (1987), 325330.

•
J.C. GómezLarrañaga, F.J. GonzálezAcuña, J. Hoste. Minimal^{} Atlases on 3manifolds, Math. Proc. Camb. Phil. Soc. 109 (1991), 105115.

•
J. Hempel. 3manifolds, Princeton University Press 1976.

•
P. Orlik. Seifert Manifolds, Lecture Notes in Math. 291, 1972 SpringerVerlag.

•
P. Scott. The geometry^{} of 3manifolds, Bull. London Math. Soc. 15 (1983), 401487.
Title  3manifold 
Canonical name  3manifold 
Date of creation  20130322 15:40:55 
Last modified on  20130322 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 