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In the previous notes, we established the Gleason-Yamabe theorem:

Theorem 1 (Gleason-Yamabe theorem) Let ${G}$ be a locally compact group. Then, for any open neighbourhood ${U}$ of the identity, there exists an open subgroup ${G'}$ of ${G}$ and a compact normal subgroup ${K}$ of ${G'}$ in ${U}$ such that ${G'/K}$ is isomorphic to a Lie group.

Roughly speaking, this theorem asserts the “mesoscopic” structure of a locally compact group (after restricting to an open subgroup ${G'}$ to remove the macroscopic structure, and quotienting out by ${K}$ to remove the microscopic structure) is always of Lie type.

In this post, we combine the Gleason-Yamabe theorem with some additional tools from point-set topology to improve the description of locally compact groups in various situations.

We first record some easy special cases of this. If the locally compact group ${G}$ has the no small subgroups property, then one can take ${K}$ to be trivial; thus ${G'}$ is Lie, which implies that ${G}$ is locally Lie and thus Lie as well. Thus the assertion that all locally compact NSS groups are Lie (Theorem 10 from Notes 4) is a special case of the Gleason-Yamabe theorem.

In a similar spirit, if the locally compact group ${G}$ is connected, then the only open subgroup ${G'}$ of ${G}$ is the full group ${G}$; in particular, by arguing as in the treatment of the compact case (Exercise 19 of Notes 3), we conclude that any connected locally compact Hausdorff group is the inverse limit of Lie groups.

Now we return to the general case, in which ${G}$ need not be connected or NSS. One slight defect of Theorem 1 is that the group ${G'}$ can depend on the open neighbourhood ${U}$. However, by using a basic result from the theory of totally disconnected groups known as van Dantzig’s theorem, one can make ${G'}$ independent of ${U}$:

Theorem 2 (Gleason-Yamabe theorem, stronger version) Let ${G}$ be a locally compact group. Then there exists an open subgoup ${G'}$ of ${G}$ such that, for any open neighbourhood ${U}$ of the identity in ${G'}$, there exists a compact normal subgroup ${K}$ of ${G'}$ in ${U}$ such that ${G'/K}$ is isomorphic to a Lie group.

We prove this theorem below the fold. As in previous notes, if ${G}$ is Hausdorff, the group ${G'}$ is thus an inverse limit of Lie groups (and if ${G}$ (and hence ${G'}$) is first countable, it is the inverse limit of a sequence of Lie groups).

It remains to analyse inverse limits of Lie groups. To do this, it helps to have some control on the dimensions of the Lie groups involved. A basic tool for this purpose is the invariance of domain theorem:

Theorem 3 (Brouwer invariance of domain theorem) Let ${U}$ be an open subset of ${{\bf R}^n}$, and let ${f: U \rightarrow {\bf R}^n}$ be a continuous injective map. Then ${f(U)}$ is also open.

We prove this theorem below the fold. It has an important corollary:

Corollary 4 (Topological invariance of dimension) If ${n > m}$, and ${U}$ is a non-empty open subset of ${{\bf R}^n}$, then there is no continuous injective mapping from ${U}$ to ${{\bf R}^m}$. In particular, ${{\bf R}^n}$ and ${{\bf R}^m}$ are not homeomorphic.

Exercise 1 (Uniqueness of dimension) Let ${X}$ be a non-empty topological space. If ${X}$ is a manifold of dimension ${d_1}$, and also a manifold of dimension ${d_2}$, show that ${d_1=d_2}$. Thus, we may define the dimension ${\hbox{dim}(X)}$ of a non-empty manifold in a well-defined manner.

If ${X, Y}$ are non-empty manifolds, and there is a continuous injection from ${X}$ to ${Y}$, show that ${\hbox{dim}(X) \leq \hbox{dim}(Y)}$.

Remark 1 Note that the analogue of the above exercise for surjections is false: the existence of a continuous surjection from one non-empty manifold ${X}$ to another ${Y}$ does not imply that ${\hbox{dim}(X) \geq \hbox{dim}(Y)}$, thanks to the existence of space-filling curves. Thus we see that invariance of domain, while intuitively plausible, is not an entirely trivial observation.

As we shall see, we can use Corollary 4 to bound the dimension of the Lie groups ${L_n}$ in an inverse limit ${G = \lim_{n \rightarrow \infty} L_n}$ by the “dimension” of the inverse limit ${G}$. Among other things, this can be used to obtain a positive resolution to Hilbert’s fifth problem:

Theorem 5 (Hilbert’s fifth problem) Every locally Euclidean group is isomorphic to a Lie group.

Again, this will be shown below the fold.

Another application of this machinery is the following variant of Hilbert’s fifth problem, which was used in Gromov’s original proof of Gromov’s theorem on groups of polynomial growth, although we will not actually need it this course:

Proposition 6 Let ${G}$ be a locally compact ${\sigma}$-compact group that acts transitively, faithfully, and continuously on a connected manifold ${X}$. Then ${G}$ is isomorphic to a Lie group.

Recall that a continuous action of a topological group ${G}$ on a topological space ${X}$ is a continuous map ${\cdot: G \times X \rightarrow X}$ which obeys the associativity law ${(gh)x = g(hx)}$ for ${g,h \in G}$ and ${x \in X}$, and the identity law ${1x = x}$ for all ${x \in X}$. The action is transitive if, for every ${x,y \in X}$, there is a ${g \in G}$ with ${gx=y}$, and faithful if, whenever ${g, h \in G}$ are distinct, one has ${gx \neq hx}$ for at least one ${x}$.

The ${\sigma}$-compact hypothesis is a technical one, and can likely be dropped, but we retain it for this discussion (as in most applications we can reduce to this case).

Exercise 2 Show that Proposition 6 implies Theorem 5.

Remark 2 It is conjectured that the transitivity hypothesis in Proposition 6 can be dropped; this is known as the Hilbert-Smith conjecture. It remains open; the key difficulty is to figure out a way to eliminate the possibility that ${G}$ is a ${p}$-adic group ${{\bf Z}_p}$. See this previous blog post for further discussion.