Content deleted Content added
m disambig link |
No edit summary |
||
| (13 intermediate revisions by 11 users not shown) | |||
Line 1:
{{no footnotes|date=June 2015}}
In [[mathematics]], in the sub-field of [[geometric topology]], the '''torus bundle''' is a kind of [[3-manifold]]. ▼
▲
==Construction==
To obtain a torus bundle: let <math>f</math> be an [[orientability|orientation]]-preserving [[homeomorphism]] of the two-dimensional [[torus]] <math>T</math> to itself. Then the three-manifold <math>M(f)</math> is obtained by
* taking the [[Cartesian product]] of <math>T</math> and the [[unit interval]] and
* gluing one component of the [[Boundary (topology)|boundary]] of the resulting manifold to the other boundary component via the map <math>f</math>.
Then <math>M(f)</math> is the torus bundle with [[monodromy]] <math>f</math>.
==Examples==
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.
Seeing the possible kinds of torus bundles in more detail requires an understanding of [[William Thurston]]'s [[Thurston's geometrization conjecture|geometrization]] program. Briefly, if <math>f</math> is [[glossary of group theory|finite order]], then the manifold <math>M(f)</math> has [[Euclidean geometry]]. If <math>f</math> is a power of a [[Dehn twist]] then <math>M(f)</math> has [[Nil geometry]]. Finally, if <math>f</math> is an [[Anosov map]] then the resulting three-manifold has [[Sol geometry]].
These three cases exactly correspond to the three possibilities for the absolute value of the trace of the action of <math>f</math> on the [[homology (mathematics)|homology]] of the torus: either less than two, equal to two, or greater than two.
==References==
*{{cite book |author=Jeffrey R. Weeks |title=The Shape of Space |url=https://archive.org/details/shapeofspace0000week |url-access=registration |year=2002 |publisher=Marcel Dekker, Inc. |edition=Second |ISBN=978-0824707095}}
[[Category:Fiber bundles]]
[[Category:Geometric topology]]
[[Category:3-manifolds]]
| |||