Solenoid question

Shadow · **Posted:** Tue, 3 Apr 2012 05:52:19 UTC

If I have the exact sequence $0\to\mathbb{Z}_p\to S_p \to \mathbb{R}/\mathbb{Z}\to 0$ , where $S_p=\varprojlim_{n\ge 0}\mathbb{R}/p^n\mathbb{Z}$ is the projective limit indicated with the obvious projection maps, then do I know that $f^{-1}(m^{-1}\mathbb{Z}/\mathbb{Z})$ is isomorphic to $C_m\times\mathbb{Z}_p$ ? (Here

is the canonical projection map to $\mathbb{R}/\mathbb{Z}$ which is the one furnished by the definition of S_p

as an inverse limit to the initial bit $\mathbb{R}/\mathbb{Z}$ )?

I feel like the answer must be something like "yes, because the fiber of a discrete group is just that group crossed with the fiber", but I've never done this before, and I don't see why the set structure which I definitely believe is a set of cardinality m crossed with $\mathbb{Z}_p$ should be inherited in the group structure.

Really I'm looking for just a bit of elaboration on how the topology and the algebra are interacting here and what gives the result if it is true, and why it is false if it is false, it doesn't seem hard, but I need to know why it works if I'm going to understand it or use the idea in the future. Thanks, and any help appreciated.

outermeasure · **Posted:** Tue, 3 Apr 2012 10:00:53 UTC

Shadow wrote:

If I have the exact sequence $0\to\mathbb{Z}_p\to S_p \to \mathbb{R}/\mathbb{Z}\to 0$ , where $S_p=\varprojlim_{n\ge 0}\mathbb{R}/p^n\mathbb{Z}$ is the projective limit indicated with the obvious projection maps, then do I know that $f^{-1}(m^{-1}\mathbb{Z}/\mathbb{Z})$ is isomorphic to $C_m\times\mathbb{Z}_p$ ? (Here

is the canonical projection map to $\mathbb{R}/\mathbb{Z}$ which is the one furnished by the definition of S_p

as an inverse limit to the initial bit $\mathbb{R}/\mathbb{Z}$ )?

I feel like the answer must be something like "yes, because the fiber of a discrete group is just that group crossed with the fiber", but I've never done this before, and I don't see why the set structure which I definitely believe is a set of cardinality m crossed with $\mathbb{Z}_p$ should be inherited in the group structure.

Really I'm looking for just a bit of elaboration on how the topology and the algebra are interacting here and what gives the result if it is true, and why it is false if it is false, it doesn't seem hard, but I need to know why it works if I'm going to understand it or use the idea in the future. Thanks, and any help appreciated.

Well, I trust you have seen the fact that the p-adic solenoid $\mathbb{S}_p$ has a unique cyclic subgroup of order m, for every $m\geq 1$ prime to p. That will give you a splitting
$0\to\mathbb{Z}_p\to f^{-1}(m^{-1}\mathbb{Z}/\mathbb{Z})\rightleftharpoons \underbrace{C_m}_{\subset\mathbb{R}/\mathbb{Z}}\to 0$
if you check the construction (in $\mathbb{R}/p^n\mathbb{Z}$ the nonidentity elements of C_m

don't live in $\mathbb{Z}/p^n\mathbb{Z}$ ).

On the other hand, when $p\mid m$ you cannot have such a splitting --- the p-adic solenoid has no

-torsion (the composition $\mathbb{Z}/p\mathbb{Z}\to\mathbb{S}_p\to\mathbb{R}/p^{n+1}\mathbb{Z}\to\mathbb{R}/p^n\mathbb{Z}$ must be trivial).

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Solenoid question

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