3-manifold3Manifolds2006-02-15 17:27:212009-11-20 01:36:08DefinitionSeifert fiber spacesbundle% this is the default PlanetMath preamble. as your knowledge
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%\newcommand{\mathbb}[1]{\mathbbmss{#1}}In this brief note we define and give instances of the notion of a 3-manifold.
A \emph{3-manifold} 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:
\begin{enumerate}
\item For example, with the Cartesian product we can get:
\begin{itemize}
\item $S^2\times S^1$
\item ${\mathbb{R}}P^2\times S^1$
\item $T\times S^1$
\item $K\times S^1$
\end{itemize}
where $S^1$ and $S^2$ are the 1- and 2-dimensional spheres respectively, $T$ is a torus, $K$ a Klein bottle, and $\mathbb{R}P^2$ is the 2-dimensional real projective space.
\item Also by the generalization of the Cartesian product: \emph{fiber bundles}, one can build bundles $E$ of the type
$$F\subset E\to S^1$$
where $F$ is any closed surface.
\item Or interchanging the roles, bundles as:
$$S^1\subset E\to F$$
\item knots and links complements
\end{enumerate}
For the second type it is known that for each \emph{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 \emph{monodromy} for $E_{\phi}$.
From the previous paragraph we infer that the \emph{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 \emph{orbifold} instead of a simple surface to get a class of 3-manifolds called \emph{Seifert fiber spaces} which are a large class of spaces needed to understand the modern classifications for 3-manifolds.
{\bf References}
\begin{itemize}
\item J.C. G\'omez-Larra\~naga. {\it 3-manifolds which are unions of three solid tori},
Manuscripta Math. 59 (1987), 325-330.
\item J.C. G\'omez-Larra\~naga, F.J. Gonz\'alez-Acu\~na, J. Hoste.
{\it Minimal Atlases on 3-manifolds},
Math. Proc. Camb. Phil. Soc. 109 (1991), 105-115.
\item J. Hempel. {\it 3-manifolds}, Princeton University Press 1976.
\item P. Orlik. {\it Seifert Manifolds}, Lecture Notes in Math. 291,
1972 Springer-Verlag.
\item P. Scott. {\it The geometry of 3-manifolds}, Bull. London Math. Soc. 15 (1983), 401-487.
\end{itemize}