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1	Representable Functor
1	Circle
1	Geometry
1	Matrix
1	Homeomorphism
1	Topological Space
1	Basic Polynomial
1	Interval
1	Simple Algebraic System
1	Class
1	Mapping
1	Elementarily Equivalent
1	Integer
1	Generating Set Of A Group
1	Group
2	Variety
2	Unit
2	Surface
2	Algebraic Topology
2	Valency
3	Orientation
3	Quotient Space
3	Neighborhood Of A Vertex
3	Generator
4	Fundamental Group
4	Fibre
4	Flow
4	Periodic Group
5	Torus
5	Period
6	Fiber Bundle
8	Manifolds
10	Surface Of Revolution
11	Isotopy
11	Monodromy
12	Orbifold
15	Mapping Class Group
17	Seifert Fiber Space
23	H N N Extension

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Surface Bundle Over The Circle

A surface bundle over is a closed 3-manifold which is constructed as a fiber bundle over the circle with fiber a closed surface.

$\displaystyle F\subset E\to S^1.$

This construction it is also a particular case of a more general concept called mapping torus.

The precise construction is as follows: Take any surface and multiply by the unit interval to get $F\times I$ . Choose any $\phi$ autohomeomorphism of . Then the quotient space

$\displaystyle E_{\phi}=\frac{F\times I}{(x,0)\sim(\phi(x),1)}$

defines a 3-manifold, characterized by the isotopy class of $\phi$ , that is, any other representative of the same class is going to produce a bundle homeomorphic to the original one. The isotopy class is called the monodromy for the bundle. It is also used for $E_{\phi}$ the symbol:

$\displaystyle F\times _{\phi}S^1$

This construction is an important source of examples in low dimensional topology as well in geometric group theory, because the geometry associated to the monodromy's action and because the bundle's fundamental group can be viewed as a particular kind of HNN extension: the fundamenal group of extended by the integers. More precisely, if $\pi^*=g_1g_2g_1^{-1}g_2^{-1}\cdots g_{k-1}g_{k}g_{k-1}^{-1}g_{k}^{-1}$ or $\pi^*=g_1^2\cdots g_k^2$ then

$\displaystyle \pi_1(E_{\phi})=\langle g_1,...,g_k,x\ \vert\ \pi^*=1, xg_k x^{-1}=\phi_*(g_k)\rangle,$

depending on

is orientable or non-orientable.

When one considers periodic monodromies it is an amusing situation since, in this case, the bundles can be seen as Seifert fiber space i.e. bundles of the form

$\displaystyle S^1\subset E\to G$

where

may be, perhaps, an orbifold.

For example, it is known that the extended mapping class group of the torus is $GL_2({\mathbb{Z}})$ , so there are only seven periodic elements, corresponding to seven Seifert fiber space already studied by J.Hempel.

Seven torus bundles $T\subset M\to S^1$ .

It is known that the following matrices generate ${\cal{M}}^*(T)$

$\displaystyle t_a =\left(\begin{array}{cc} 1 & -1 \\ 0 & 1 \\ \end{array}\right) ,\quad t_b =\left(\begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array}\right)$ and $\displaystyle \quad y= \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array}\right)$

which obey

$\displaystyle \langle t_a,t_b,y\ :\ (t_at_bt_a)^4=1,t_at_bt_a=t_bt_at_b,y^2=1,yt_ay^{-1}={t_a}^{-1}, yt_by^{-1}={t_b}^{-1}\rangle$

The first two are left twists from a simple meridian curve and a simple longitude curve. The matrix for represents a autohomeomorphism which is not a twist and inverts orientation. It is obtained by inverting curve 's direction and extending in a regular neighborhood , then extending to $N(a\cup b)$ and finally in a disk to the whole torus.

Now we can represent the periodic monodromies of example 12.4 in [Hempel, pp.122-123] in terms of those generators as

$\displaystyle 1= \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array}\right)... ...ight) ,\ t_ay= \left(\begin{array}{cc} 1 & 1 \\ 0 & -1 \\ \end{array}\right)$

$\displaystyle (t_bt_a)^2= \left(\begin{array}{cc} 0 & -1 \\ 1 & -1 \\ \end{ar... ...ht) ,\ t_at_b= \left(\begin{array}{cc} 0 & -1 \\ 1 & 1 \\ \end{array}\right)$

of periods 1, 2, 2, 2, 3, 4, 6 respectively [Hempel, pp.123].

And in turn give the Seifert fiber spaces

$(Oo,1\vert\ 0)=T\times S^1$

$(On,2\vert\ 0)=(Oo,0\vert\ (1,-2),(2,1),(2,1),(2,1),(2,1))$

$(No,1\vert\ 0)=(NnI,2\vert\ 0)=K\times S^1$

$(No,1\vert\ 1)=(NnI,2\vert\ 1))=K\times_{\tau} S^1$

$(Oo,0\vert\ (3,2),(3,-1),(3,-1))$

$(Oo,0\vert\ (2,1),(4,-1),(4,-1))$

$(Oo,0\vert\ (2,1),(3,-1),(6,-1))$

References:

J. Hempel, 3-manifolds, Annals of Math. Studies, 86, Princeton Univ. Press 1976.
P. Orlik, Seifert Manifolds, Lecture Notes in Math. 291, 1972 Springer-Verlag.
P. Orlik, F. Raymond, On 3-manifolds with local ${\rm SO}_2$ action, Quart. J. Math. Oxford Ser.(2) 20 (1969), 143-160.
H. Seifert, Topologie dreidimensionaler gefaserter Räume, 60(1933), 147-238.