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| | My name: Katherin Sansom<br>My age: 38 years old<br>Country: France<br>Home town: Sarreguemines <br>ZIP: 57200<br>Address: 21 Rue Des Six Freres Ruellan<br><br>Feel free to surf to my homepage [http://tinyurl.com/lhambsd nike sko] |
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| In [[topology]], a branch of mathematics, '''irrational winding of a torus''' is a continuous injection of a line into a torus that is used to set up several counterexamples.<ref>{{cite book|url=http://books.google.com/books?id=ILhUYVmvHt0C&pg=PA45|title=Compact Lie groups and their representations|author=D. P. Zhelobenko}}</ref> A related notion is [[Kronecker foliation]].
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| == Definition ==
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| Consider a [[torus]] <math>T^2 = \mathbb{R}^2 / \mathbb{Z}^2</math>, and the corresponding [[quotient space|projection]] <math>\pi: \mathbb{R}^2 \to T^2</math>. The points of the torus correspond to (translated) points of a square lattice in <math>\mathbb{R}^2</math> that is <math>\mathbb{Z}^2</math>, and <math>\pi</math> factors through a map that takes any point to a point in <math>[0, 1)^2</math> given by the fractional parts of the original point's standard coordinates. Now consider a line in <math>\mathbb{R}^2</math> given by the equation ''y = kx''. If ''k'' is [[rational number|rational]], then it can be represented by a fraction and a corresponding lattice point of <math>\mathbb{Z}^2</math>. It can be shown that then the projection of this line is a [[simple curve|simple]] [[closed curve]] on a torus. If, however, ''k'' is [[irrational number|irrational]], then it will not cross any lattice points except 0, which means that its projection on the torus will not be a closed curve, and the restriction of <math>\pi</math> on this line is [[injective mapping|injective]]. Moreover, it can be shown that the image of this restricted projection as a subspace, called the irrational winding of a torus, is [[dense subspace|dense]] in the torus.
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| == Applications ==
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| Irrational winding of a torus is used to set up a few counter-examples related to [[monomorphism]]s. It is an [[Submanifold#Immersed_submanifolds|immersed submanifold]] but not a [[Submanifold#Embedded_submanifolds|regular submanifold]] of the torus, which shows that the image of a manifold under a [[continuous function|continuous]] injection to another manifold is not necessarily a (regular) submanifold.<ref name="Tu">{{cite book|author=Loring W. Tu |title=An Introduction to Manifolds | publisher=Springer |year=2010| pages=168 | isbn=978-1-4419-7399-3}}
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| </ref> It is also an example of the fact that the induced submanifold topology does not have to coincide with the [[subspace topology]] of the submanifold <ref name="Tu"/> {{cref|a}}
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| Secondly, the torus can be considered as a [[Lie group]] <math>U(1) \times U(1)</math>, and the line can be considered as <math>\mathbb{R}</math>. Then it is easy to show that the image of the continuous and analytic [[group homomorphism]] <math>x \mapsto (e^{ix}, e^{ikx})</math> is not a Lie subgroup<ref name="Tu"/><ref>{{Citation | last1=Čap |first1=Andreas | authorlink=Andreas Čap | last2=Slovák | first2=Jan |
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| title=Parabolic Geometries: Background and general theory | publisher=AMS | year=2009 | url=http://books.google.com/books/about/Parabolic_Geometries_Background_and_gene.html?id=G4Ot397nWsQC | isbn=978-0-8218-2681-2 | pages=24}}
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| </ref> (because it's not closed in the torus) while, of course, it is still a group. It is also used to show that if a subgroup ''H'' of the Lie group ''G'' is not closed, the quotient ''G/H'' does not need to be a submanifold<ref>{{citation | first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997 | isbn = 0-387-94732-9 | pages=146}}</ref> and even not a [[Hausdorff space]].
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| == See also ==
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| * [[Torus knot]]
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| == Notes ==
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| {{cnote|a|As a topological [[Subspace topology|subspace]] of the torus, the irrational winding is not a [[manifold]] at all, because it is not locally homeomorphic to <math>\mathbb{R}</math>.}}
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| == References ==
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| <references/>
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| [[Category:General topology]]
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| [[Category:Lie groups]]
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| [[Category:Topological spaces]]
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My name: Katherin Sansom
My age: 38 years old
Country: France
Home town: Sarreguemines
ZIP: 57200
Address: 21 Rue Des Six Freres Ruellan
Feel free to surf to my homepage nike sko