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| In [[mathematics]], in the sub-field of [[geometric topology]], a '''torus bundle''' is a kind of [[surface bundle over the circle]], which in turn are a class of [[three-manifold]]s.
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| ==Construction==
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| To obtain a '''torus bundle''': let <math>f</math> be an
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| [[orientability|orientation]]-preserving [[homeomorphism]] of the
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| two-dimensional [[torus]] <math>T</math> to itself.
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| Then the three-manifold <math>M(f)</math> is obtained by
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| * taking the [[Cartesian product]] of <math>T</math> and the [[unit interval]] and
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| * gluing one component of the [[Boundary (topology)|boundary]] of the resulting manifold to the other boundary component via the map <math>f</math>.
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| Then <math>M(f)</math> is the torus bundle with [[monodromy]] <math>f</math>.
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| ==Examples==
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| For example, if <math>f</math> is the identity map (i.e., the map which fixes every point of the torus) then the resulting torus bundle <math>M(f)</math> is the [[three-torus]]: the Cartesian product of three [[circle]]s.
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| Seeing the possible kinds of torus bundles in more detail
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| requires an understanding of [[William Thurston]]'s
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| [[Thurston's geometrization conjecture|geometrization]] program.
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| Briefly, if <math>f</math> is [[glossary of group theory|finite order]],
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| then the manifold <math>M(f)</math> has [[Euclidean geometry]].
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| If <math>f</math> is a power of a [[Dehn twist]] then <math>M(f)</math> has
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| [[Nil geometry]]. Finally, if <math>f</math> is an [[Anosov map]] then the
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| resulting three-manifold has [[Sol geometry]].
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| These three cases exactly correspond to the three possibilities
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| for the absolute value of the trace of the action of <math>f</math> on the
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| [[homology (mathematics)|homology]] of the torus: either less than two, equal to two,
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| or greater than two.
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| ==References==
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| Anyone seeking more information on this subject, presented
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| in an elementary way, may consult [[Jeffrey Weeks (mathematician)|Jeff Weeks]]' book
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| [[The Shape of Space]].
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| [[Category:Fiber bundles]]
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| [[Category:Geometric topology]]
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| [[Category:3-manifolds]]
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