Irrational winding of a torus

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.[1] A related notion is Kronecker foliation.

Contents

Definition

Consider a torus T^2 = \mathbb{R}^2 / \mathbb{Z}^2, and the corresponding projection \pi: \mathbb{R}^2 \to T^2. The points of the torus correspond to (translated) points of a square lattice in \mathbb{R}^2 that is \mathbb{Z}^2, and \pi factors through a map that takes any point to a point in [0, 1)^2 given by the fractional parts of the original point's standard coordinates. Now consider a line in \mathbb{R}^2 given by the equation y = kx. If k is rational, then it can be represented by a fraction and a corresponding lattice point of \mathbb{Z}^2. It can be shown that then the projection of this line is a simple closed curve on a torus. If, however, k is 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 \pi on this line is 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 in the torus.

Applications

Irrational winding of a torus is used to set up a few counter-examples related to monomorphisms. It is an immersed submanifold but not a regular submanifold of the torus, which shows that the image of a manifold under a continuous injection to another manifold is not necessarily a (regular) submanifold.[2] 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 [2] a[›]

Secondly, the torus can be considered as a Lie group U(1) \times U(1), and the line can be considered as \mathbb{R}. Then it is easy to show that the image of the continuous and analytic group homomorphism x \mapsto (e^{ix}, e^{ikx}) is not a Lie subgroup[2][3] (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[4] and even not a Hausdorff space.

See also

Notes

^ a: As a topological subspace of the torus, the irrational winding is not a manifold at all, because it is not locally homeomorphic to \mathbb{R}.

References

  1. ^ D. P. Zhelobenko. Compact Lie groups and their representations. http://books.google.com/books?id=ILhUYVmvHt0C&pg=PA45. 
  2. ^ a b c Loring W. Tu (2010). An Introduction to Manifolds. Springer. pp. 168. ISBN 978-1441973993. 
  3. ^ Čap, Andreas; Slovák, Jan (2009), Parabolic Geometries: Background and general theory, AMS, pp. 24, ISBN 978-0821826812, http://books.google.com/books/about/Parabolic_Geometries_Background_and_gene.html?id=G4Ot397nWsQC 
  4. ^ Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, pp. 146, ISBN 0-387-94732-9