Mapping torus

In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f. Take the cartesian product of X with a closed interval I, and glue the boundary components together by the static homeomorphism:

[math]\displaystyle{ M_f =\frac{(I \times X)}{(1,x)\sim (0,f(x))} }[/math]

The result is a fiber bundle whose base is a circle and whose fiber is the original space X.

If X is a manifold, M_f will be a manifold of dimension one higher, and it is said to "fiber over the circle".

As a simple example, let [math]\displaystyle{ X }[/math] be the circle, and [math]\displaystyle{ f }[/math] be the inversion [math]\displaystyle{ e^{ix} \mapsto e^{-ix} }[/math], then the mapping torus is the Klein bottle.

Mapping tori of surface homeomorphisms play a key role in the theory of 3-manifolds and have been intensely studied. If S is a closed surface of genus g ≥ 2 and if f is a self-homeomorphism of S, the mapping torus M_f is a closed 3-manifold that fibers over the circle with fiber S. A deep result of Thurston states that in this case the 3-manifold M_f is hyperbolic if and only if f is a pseudo-Anosov homeomorphism of S.^[1]

References

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