1.1. Hilbert’s fifth problem 11

(1) (Group-like object) G is a topological group.

(2) (Weak regularity) G is locally Euclidean.

Then

(i) (Lie-type structure) G is isomorphic to a Lie group.

Exercise 1.1.2. Show that a locally Euclidean topological group is neces-

sarily Hausdorff (without invoking Theorem 1.1.9).

We will prove this theorem in Section 6. As it turns out, Theorem 1.1.9

is not directly useful for many applications, because it is often diﬃcult to

verify that a given topological group is locally Euclidean. On the other

hand, the weaker property of local compactness, which is clearly implied by

the locally Euclidean property, is much easier to verify in practice. One can

then ask the more general question of whether every locally compact group

is isomorphic to a Lie group. Unfortunately, the answer to this question is

easily seen to be no, as the following examples show:

Example 1.1.10 (Trivial topology). A group equipped with the trivial

topology is a compact (hence locally compact) group, but will not be Haus-

dorff (and thus not Lie) unless the group is also trivial. Of course, this is

a rather degenerate counterexample and can be easily eliminated in prac-

tice. For instance, we will see later that any topological group can be made

Hausdorff by quotienting out the closure of the identity.

Example 1.1.11 (Infinite-dimensional torus). The infinite-dimensional

torus

(R/Z)N

(with the product topology) is an (additive) topological group,

which is compact (and thus locally compact) by Tychonoff’s theorem. How-

ever, it is not a Lie group.

Example 1.1.12 (p-adics). Let p be a prime. We define the p-adic norm

p

on the integers Z by defining n

p

:=

p−j,

where

pj

is the largest power

of p that divides n (with the convention 0

p

:= 0). This is easily verified to

generate a metric (and even an ultrametric) on Z; the p-adic integers Zp are

then defined as the metric completion of Z under this metric. This is easily

seen to be a compact (hence locally compact) additive group (topologically,

it is homeomorphic to a Cantor set). However, it is not locally Euclidean

(or even locally connected), and so is not isomorphic to a Lie group.

One can also extend the p-adic norm to the ring Z[

1

p

] of rationals of the

form

a/pj

for some integers a, j in the obvious manner; the metric com-

pletion of this space is then the p-adic rationals Qp. This is now a locally

compact additive group rather than a compact one (Zp is a compact open

neighbourhood of the identity); it is still not locally connected, so it is still

not a Lie group.