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- 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.

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,

$bigcap\_\{ige\; 0\}alpha^iT$ 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.

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Last updated on Monday September 24, 2007 at 10:08:30 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday September 24, 2007 at 10:08:30 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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