## Totally disconnected homogeneous spaces

I was hoping to give the proof of Rodier’s Mackey-type theorem in this post (the one that implies the Stone-von Neumann theorem), but in the interest of keeping this at a reasonable length I’m first going to prove a fact which I think was used implicitly in Rodier’s proof. Even if we don’t directly use this, it seems more or less essential to understanding the quasi-inverse of the equivalence in the theorem.

This result is also pretty interesting on its own. The case where is compact, hence profinite, is at the beginning of Serre’s *Galois Cohomology*.

**Lemma** If is an -group and is a closed subgroup of , then is a locally trivial -bundle. This means that any point has an open neighborhood such that there is an -equivariant homeomorphism which commutes with and the natural projection .

*Proof.* First, we claim that if is compact, i.e. profinite, then is a *globally* trivial -bundle. More generally, if are closed subgroups, then the projection admits a continuous section . To see why this implies the previous statement, take and consider the map which sends , which is clearly a continuous -equivariant bijection and commutes with the projections to . But a continuous bijection between compact Hausdorff spaces is a homeomorphism.

To prove the existence of this continuous section, assume for the moment that is finite, so there is an open subgroup of such that . This means is injective, and a continuous bijection between compact Hausdorff spaces is a homeomorphism. So has a continuous section over , and since is covered by the open translates we can put these together to obtain a continuous section over all of . Now if is not necessarily finite, consider the set of pairs where is a closed subgroup and is a continuous section. There is a natural partial ordering on these pairs: let provided that and commute with the natural projection . Obviously this partially ordered set is nonempty, since we can take and to be the identity. If is a chain of such pairs, then it is not hard to see that if is the intersection of all the subgroups and is the restriction of the sections to , then is a lower bound for , since the natural map

is a homeomorphism. Thus by Zorn’s lemma there is a minimal element , and we will show that . Otherwise, we can find an open subgroup of such that is strictly contained in , and since is finite the first argument yields a continuous section . But then , which contradicts the minimality of .

Finally, let be any -group and a closed subgroup. Clearly it suffices to show that is a trivial -bundle over some neighborhood of . For this, choose a compact open subgroup and put . By the compact case there is a continuous section , so the map is a continuous -equivariant bijection which commutes with the projections to . But this map is the disjoint union of the maps for various , and these are all homeomorphisms by the same argument as above. Since the are open in , it follows that is a homeomorphism.