Mapping torus
title: "Mapping torus" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["general-topology", "geometric-topology", "homeomorphisms"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Mapping_torus" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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, Mf 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 Mf 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 Mf is hyperbolic if and only if f is a pseudo-Anosov homeomorphism of S.
References
References
- W. Thurston, ''On the geometry and dynamics of diffeomorphisms of surfaces'', [[Bulletin of the American Mathematical Society]], vol. 19 (1988), pp. 417–431
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