surface bundle over the circle
This construction it is also a particular case of a more general concept called mapping torus.
defines a 3-manifold, characterized by the isotopy class of , 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 the symbol:
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 or then
depending on is orientable or non-orientable.
where may be, perhaps, an orbifold.
For example, it is known that the extended mapping class group of the torus is , so there are only seven periodic elements, corresponding to seven Seifert fiber space already studied by J.Hempel.
Seven torus bundles .
It is known that the following matrices generate
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 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
of periods 1, 2, 2, 2, 3, 4, 6 respectively [Hempel, pp.123].
And in turn give the Seifert fiber spaces
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 action, Quart. J. Math. Oxford Ser.(2) 20 (1969), 143-160.
H. Seifert, Topologie dreidimensionaler gefaserter Räume, 60(1933), 147-238.
|Title||surface bundle over the circle|
|Date of creation||2013-03-22 15:42:37|
|Last modified on||2013-03-22 15:42:37|
|Last modified by||juanman (12619)|