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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: : 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". 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''.〔W. Thurston, ''On the geometry and dynamics of diffeomorphisms of surfaces'', Bulletin of the American Mathematical Society, vol. 19 (1988), pp. 417–431〕 ==References== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mapping torus」の詳細全文を読む スポンサード リンク
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