You are currently browsing the tag archive for the ‘manifolds’ tag.
In the previous notes, we established the Gleason-Yamabe theorem:
Theorem 1 (Gleason-Yamabe theorem) Let
be a locally compact group. Then, for any open neighbourhood
of the identity, there exists an open subgroup
of
and a compact normal subgroup
of
in
such that
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 to remove the macroscopic structure, and quotienting out by
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 has the no small subgroups property, then one can take
to be trivial; thus
is Lie, which implies that
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 is connected, then the only open subgroup
of
is the full group
; 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 need not be connected or NSS. One slight defect of Theorem 1 is that the group
can depend on the open neighbourhood
. However, by using a basic result from the theory of totally disconnected groups known as van Dantzig’s theorem, one can make
independent of
:
Theorem 2 (Gleason-Yamabe theorem, stronger version) Let
be a locally compact group. Then there exists an open subgoup
of
such that, for any open neighbourhood
of the identity in
, there exists a compact normal subgroup
of
in
such that
is isomorphic to a Lie group.
We prove this theorem below the fold. As in previous notes, if is Hausdorff, the group
is thus an inverse limit of Lie groups (and if
(and hence
) 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
be an open subset of
, and let
be a continuous injective map. Then
is also open.
We prove this theorem below the fold. It has an important corollary:
Corollary 4 (Topological invariance of dimension) If
, and
is a non-empty open subset of
, then there is no continuous injective mapping from
to
. In particular,
and
are not homeomorphic.
Exercise 1 (Uniqueness of dimension) Let
be a non-empty topological space. If
is a manifold of dimension
, and also a manifold of dimension
, show that
. Thus, we may define the dimension
of a non-empty manifold in a well-defined manner.
If
are non-empty manifolds, and there is a continuous injection from
to
, show that
.
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
to another
does not imply that
, 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 in an inverse limit
by the “dimension” of the inverse limit
. 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
be a locally compact
-compact group that acts transitively, faithfully, and continuously on a connected manifold
. Then
is isomorphic to a Lie group.
Recall that a continuous action of a topological group on a topological space
is a continuous map
which obeys the associativity law
for
and
, and the identity law
for all
. The action is transitive if, for every
, there is a
with
, and faithful if, whenever
are distinct, one has
for at least one
.
The -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).
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
is a
-adic group
. See this previous blog post for further discussion.
Recent Comments