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.
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Consider a torus , and the corresponding projection . The points of the torus correspond to (translated) points of a square lattice in that is , and factors through a map that takes any point to a point in given by the fractional parts of the original point's standard coordinates. Now consider a line in given by the equation y = kx. If k is rational, then it can be represented by a fraction and a corresponding lattice point of . 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 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.
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 , and the line can be considered as . Then it is easy to show that the image of the continuous and analytic group homomorphism 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.
^ a: As a topological subspace of the torus, the irrational winding is not a manifold at all, because it is not locally homeomorphic to .