Solenoid (mathematics)

This page discusses the class of topological group. For the wrapped loop of wire, see Solenoid.

In mathematics, for a given prime number p, the p-adic solenoid is the topological group defined as inverse limit of the inverse system

(S_i, q_i)

where i runs over natural numbers, and each S_i is a circle, and q_i wraps the circle S_i+1 p times around the circle S_i.

The solenoid is the standard example of a space with bad behaviour with respect to various homology theories, not seen for simplicial complexes. For example, in Čech homology, one can construct a non-exact long homology sequence using the solenoid. In Steenrod-style homology theories the 0th homology group of the solenoid tends to have a fairly complicated structure, even though the solenoid is a connected space.

Embedding in R³

The solid torus αT embedded inside T, for the case p=3

An embedding of the p-adic solenoid into R³ can be constructed in the following way. Take a solid torus T in R³ and choose an embedding α: T → T such that α acts on the fundamental group of T as multiplication by p; that is to say, α maps T onto a solid torus inside T which winds p times around the axis of T before joining up with itself. Then the ω-limit set of α, that is,

$\bigcap _{i\geq 0}\alpha ^{i}T$ the intersection (in R³) of the smaller and smaller toruses T, αT, α(αT), etc., is a p-adic solenoid inside T, hence in R³.

One way to see that this is true involves seeing that this set is the inverse limit of the inverse system consisting of infinitely many copies of T with maps α between them, and this system is topologically equivalent to the inverse system (S_i, q _i) defined above.

This construction shows how the p-adic solenoid arises in the study of dynamical systems on R³ (since α can arise as the restriction of a continuous map R³ → R³). It is an example of a nontrivial indecomposable continuum.