# Irrational winding of a torus

(Redirected from Irrational cable on 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 ${\displaystyle T^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection ${\displaystyle \pi :\mathbb {R} ^{2}\to T^{2}}$. Each point in the torus has as its preimage one of the translates of the square lattice ${\displaystyle \mathbb {Z} ^{2}}$ in ${\displaystyle \mathbb {R} ^{2}}$, and ${\displaystyle \pi }$ factors through a map that takes any point in the plane to a point in the unit square ${\displaystyle [0,1)^{2}}$ given by the fractional parts of the original point's Cartesian coordinates. Now consider a line in ${\displaystyle \mathbb {R} ^{2}}$ 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 ${\displaystyle \mathbb {Z} ^{2}}$. 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 ${\displaystyle \pi }$ 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 ${\displaystyle U(1)\times U(1)}$, and the line can be considered as ${\displaystyle \mathbb {R} }$. Then it is easy to show that the image of the continuous and analytic group homomorphism ${\displaystyle x\mapsto (e^{ix},e^{ikx})}$ 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 ${\displaystyle \mathbb {R} }$.