surface bundle over the circle

A bundle over S1 is a closed 3-manifold which is constructed as a fiber bundleMathworldPlanetmath over the circle with fiber a closed surface.


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 F and multiply by the unit interval I to get F×I. Choose any ϕ autohomeomorphism of F. Then the quotient spaceMathworldPlanetmath


defines a 3-manifold, characterized by the isotopyMathworldPlanetmathPlanetmath class of ϕ, 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 Eϕ the symbol:


This construction is an important source of examples in low dimensional topologyMathworldPlanetmath as well in geometric group theory, because the geometry associated to the monodromy’s action and because the bundle’s fundamental groupMathworldPlanetmathPlanetmath can be viewed as a particular kind of HNN extension: the fundamenal group of F extended by the integers. More precisely, if π*=g1g2g1-1g2-1gk-1gkgk-1-1gk-1 or π*=g12gk2 then


depending on F 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


where G may be, perhaps, an orbifoldMathworldPlanetmath.

For example, it is known that the extended mapping class groupPlanetmathPlanetmath of the torus is GL2(), so there are only seven periodic elements, corresponding to seven Seifert fiber space already studied by J.Hempel.

Seven torus bundles TMS1.

It is known that the following matrices generate *(T)


which obey


The first two are left twists from a a simple meridian curve and b a simple longitude curve. The matrix for y represents a autohomeomorphism which is not a twist and inverts orientation. It is obtained by inverting curve b’s direction and extending in a regularPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath neighborhoodMathworldPlanetmathPlanetmath N(b), then extending to N(ab) 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 generatorsPlanetmathPlanetmathPlanetmath as


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

And in turn give the Seifert fiber spaces

(Oo,1| 0)=T×S1

(On,2| 0)=(Oo,0|(1,-2),(2,1),(2,1),(2,1),(2,1))

(No,1| 0)=(NnI,2| 0)=K×S1

(No,1| 1)=(NnI,2| 1))=K×τS1





  1. 1.

    J. Hempel, 3-manifolds, Annals of Math. Studies, 86, Princeton Univ. Press 1976.

  2. 2.

    P. Orlik, Seifert Manifolds, Lecture Notes in Math. 291, 1972 Springer-Verlag.

  3. 3.

    P. Orlik, F. Raymond, On 3-manifolds with local SO2 action, Quart. J. Math. Oxford Ser.(2) 20 (1969), 143-160.

  4. 4.

    H. Seifert, Topologie dreidimensionaler gefaserter Räume, 60(1933), 147-238.

Title surface bundle over the circle
Canonical name SurfaceBundleOverTheCircle
Date of creation 2013-03-22 15:42:37
Last modified on 2013-03-22 15:42:37
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 24
Author juanman (12619)
Entry type Definition
Classification msc 55R10
Classification msc 57M50
Classification msc 57N10
Related topic FiberBundle
Related topic FourSurfaceBundles