# 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.

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