Lesson plan
Mapping torus
In mathematics, specifically in topology, the mapping torus 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: M_f =\frac{(I \times X)}{(1,x)\sim (0,f(x))} 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 X be the circle, and f be the inversion e^{ix} \mapsto e^{-ix} , 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.
Article summary
In mathematics, specifically in topology, the mapping torus 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: M_f =\frac{(I \times X)}{(1,x)\sim (0,f(x))} 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 X be the circle, and f be the inversion e^{ix} \mapsto e^{-ix} , 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.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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