Can we hope for application of Etale cohomology techniques in proving results concerning semialgebraic subsets of $\mathbb{Q}\_p^n$?


Recall that semialgebraic subsets are obtained from $p$-adic algebraic varieties using boolean connectives and coordinate projection.


The results I am interested in are a sort of algebraic topology over the $p$-adics. But of course one cannot use the standard topology on the $p$-adics since it is totally disconnected. So I hope to show for instance that a ball with one smaller ball removed cannot be diffeomorphic to a ball, etc .



Etale cohomology in the $p$-adic setting
1) A ball (of finite radius) is compact, and a punctured ball is not, so they cannot be diffeomorphic.


2) A theorem of Serre (*Topology*, 1965, vol. **3**, p. 409-412) classifies all compact $p$-adic manifolds.
The result is: 



> 
> a) Any such (non-empty) manifold is isomorphic to a finite disjoint union of balls.
> 
> b) For positive integers $a$ and $b$, the union of $a$ balls is isomorphic to the union of $b$ ones if and only if $a\equiv b\pmod{p-1}$.
> 


For example, the unit ball is isomorphic to the union of $p$ disjoint balls (the residue classes).


[Edit, 6/15/2012: I corrected and added the reference to Serre's paper.]


