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

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${S}^{2}\times {S}^{1}$

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$\mathbf{R}{P}^{2}\times {S}^{1}$

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$T\times {S}^{1}$

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…

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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 3manifolds called Seifert fiber spaces which are a large class of spaces needed to understand the modern classifications for 3manifolds.

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

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

[H
] J. Hempel. 3manifolds, Princeton University Press 1976.

[O
] P. Orlik. Seifert Manifolds, Lecture Notes in Math. 291, 1972 SpringerVerlag.

[S
] P. Scott. The geometry of 3manifolds, Bull. London Math. Soc. 15 (1983), 401487.

[G
Title  3manifold 

Canonical name  3manifold 
Date of creation  20130311 19:23:28 
Last modified on  20130311 19:23:28 
Owner  juanman (12619) 
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