formulasearchengine - User contributions [en]

Relative biological effectiveness

2014-02-27T14:41:31Z

166.111.33.47: /* History */

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Euclidean division

2013-09-25T02:57:23Z

166.111.81.160: /* Generalized division algorithms */

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.

==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==

Anyone seeking more information on this subject, presented
in an elementary way, may consult [[Jeffrey Weeks (mathematician)|Jeff Weeks]]' book
[[The Shape of Space]].

[[Category:Fiber bundles]]
[[Category:Geometric topology]]
[[Category:3-manifolds]]

Quantum spin Hall effect

2012-08-14T23:45:40Z

166.111.177.133:

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