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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 G is compact, hence profinite, is at the beginning of Serre’s Galois Cohomology.

Lemma If G is an \ell-group and H is a closed subgroup of G, then \pi : G \to G/H is a locally trivial H-bundle. This means that any point x \in G/H has an open neighborhood U such that there is an H-equivariant homeomorphism \pi^{-1}(U) \to U \times H which commutes with \pi and the natural projection U \times H \to U.

Proof. First, we claim that if G is compact, i.e. profinite, then \pi : G \to G/H is a globally trivial H-bundle. More generally, if K \subset H \subset G are closed subgroups, then the projection \pi : G/K \to G/H admits a continuous section s : G/H \to G/K. To see why this implies the previous statement, take K = 1 and consider the map G/H \times H \to G which sends (x,h) \mapsto s(x)h, which is clearly a continuous H-equivariant bijection and commutes with the projections to G/H.  But a continuous bijection between compact Hausdorff spaces is a homeomorphism.

To prove the existence of this continuous section, assume for the moment that H/K is finite, so there is an open subgroup U of G such that H \cap U \subset K. This means \pi|_{UK/K} is injective, and a continuous bijection between compact Hausdorff spaces is a homeomorphism. So \pi has a continuous section over \pi(UK/K), and since G/H is covered by the open translates g \cdot \pi(UK/K) we can put these together to obtain a continuous section over all of G/H. Now if H/K is not necessarily finite, consider the set of pairs (L,s) where K \subset L \subset H is a closed subgroup and s : G/H \to G/L is a continuous section. There is a natural partial ordering on these pairs: let (L_1,s_1) \leq (L_2,s_2) provided that L_1 \subset L_2 and s_1,s_2 commute with the natural projection G/L_1 \to G/L_2. Obviously this partially ordered set is nonempty, since we can take L = H and s to be the identity. If \mathcal{C} is a chain of such pairs, then it is not hard to see that if L is the intersection of all the subgroups and s is the restriction of the sections to L, then (L,s) is a lower bound for \mathcal{C}, since the natural map

G/L \to \varprojlim_{L_i \in \mathcal{C}} G/L_i

is a homeomorphism. Thus by Zorn’s lemma there is a minimal element (L,s), and we will show that L = K. Otherwise, we can find an open subgroup U of G such that L \cap U is strictly contained in L, and since L/L \cap U is finite the first argument yields a continuous section t : G/L \cap U \to G/L. But then (L \cap U,t \circ s) < (L,s), which contradicts the minimality of (L,s).

Finally, let G be any \ell-group and H a closed subgroup. Clearly it suffices to show that \pi : G \to G/H is a trivial H-bundle over some neighborhood of \pi(1). For this, choose a compact open subgroup K \subset G and put U = \pi(K). By the compact case there is a continuous section s : U \to K \subset \pi^{-1}(U), so the map (u,h) \mapsto s(u)h is a continuous H-equivariant bijection U \times H \to \pi^{-1}(U) which commutes with the projections to U. But this map is the disjoint union of the maps U \times hH \cap K \to Kh(H \cap K) for various h \in H, and these are all homeomorphisms by the same argument as above. Since the Kh(H \cap K) are open in KH = \pi^{-1}(U), it follows that U \times H \to \pi^{-1}(U) is a homeomorphism.

\Box

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