Irrational winding of a torus

In topology, a branch of mathematics, an 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 the Kronecker foliation of a torus, a foliation formed by the set of all translates of a given irrational winding.

Definition

One way of constructing a torus is as the quotient space $T^2 = \mathbb{R}^2 / \mathbb{Z}^2$ of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection $\pi: \mathbb{R}^2 \to T^2$. Each point in the torus has as its preimage one of the translates of the square lattice $\mathbb{Z}^2$ in $\mathbb{R}^2$, and $\pi$ factors through a map that takes any point in the plane to a point in the unit square $[0, 1)^2$ given by the fractional parts of the original point's Cartesian coordinates. Now consider a line in $\mathbb{R}^2$ given by the equation y = kx. If the slope k of the line 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 windings of a torus may be used to set up counter-examples related to monomorphisms. An irrational winding 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] Irrational windings are also examples 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 – see the closed subgroup theorem) while, of course, it is still a group. It may also be 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 might even fail to be a Hausdorff space.

See also

• Torus knot

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.
2. Loring W. Tu (2010). An Introduction to Manifolds. Springer. p. 168. ISBN 978-1-4419-7399-3.
3. Čap, Andreas; Slovák, Jan (2009), Parabolic Geometries: Background and general theory, AMS, p. 24, ISBN 978-0-8218-2681-2
4. Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, p. 146, ISBN 0-387-94732-9