# Etale cohomology in the $p$-adic setting

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 .

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Your question is a bit vague... What do you mean by semialgebraic subset ? What kind of results are you interested in ? –  François Brunault Nov 22 '11 at 21:11
@François: I edited the question. Thanks for the comment. –  user16974 Nov 22 '11 at 21:16

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.]

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I know these results. Can we extend these to say that there is no diffeomorphism from $\mathbbQ_p^n$ onto $\mathbb{Q}_p^n \setminus B$ for $B$ a ball? –  user16974 Nov 23 '11 at 6:35
Aren't both of them countable infinite unions of disjoint balls? –  ACL Nov 27 '11 at 13:04
What happens if you restrict to semialgebraic diffeomorphism? –  user16974 Nov 28 '11 at 11:19
For those interested, the reference for the Serre paper given here is not correct. I believe the correct reference for this result is the paper "Classification des variétés analytiques p-adiques compactes" from Topology 3 (1965), pp. 409-412. –  Michael A Warren Jun 13 '12 at 17:37
You're right.Sorry for the misprint. I correct it. –  ACL Jun 15 '12 at 13:01