Solenoid (mathematics)

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

: (Si, q i)

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

The solenoid is the canonical example of a space with bad behaviour with respect to various homology theories, not seen for simplicial complexes. For example, in ech homology, one can construct a non-exact sequence 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 =

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 : T T such that acts on the fundamental group of T as multiplication by p ; that is to say, maps T onto a solid torus inside T which winds p times around the axis of T before joining up with itself. Then the -limit set of , that is,

igcap_{ige 0}alpha^iT the intersection (in R³) of the smaller and smaller toruses T , T , (T) , 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 between them, and this system is topologically equivalent to the inverse system (Si, q i) defined above.

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