Abelian Groups - László Fuchs [Springer]

5 Locally Compact Groups 203
5 Locally Compact Groups
Having discussed compact and linearly compact groups, it is worth while making a
few comments on the locally compact case. Besides the Pontryagin duality theory
we rely heavily on homological machinery, so a reader not familiar with the material
of Chapters 7–9 is advised to read this section after studying these chapters.
Locally Compact Extensions It is well known that a locally compact (abelian)
group is the direct sum of R n (for some integer n 0) and a group G that contains an
open compact subgroup C. Thus G fits into an exact sequence 0 ! C ! G ! A !
0 where C is compact and A is discrete. Conversely, any extension of a compact C
by a discrete A yields a well-defined locally compact group G, as the only topology
compatible with the given topologies of C and A is the one in which a base of
neighborhoods of 0 in C is also a base of neighborhoods of 0 in G. Consequently,
if C is compact and A is discrete, then the elements of Ext.A; C/ may be viewed as
the locally compact extensions of C by A (the equivalence of extensions is the same
whether C is discrete or compact).
Recall that the Pontryagin dual (or character group) CharG of a locally
compact group G, i.e., the group of all continuous homomorphisms of G into the
compact group T D R=Z, equipped with the compact-open topology, isagain
locally compact. We write
OG D Char G D Hom.G; T/;
where Hom now stands for continuous homomorphisms. Importantly, if G is
compact, then OG is discrete, and vice versa. The Pontryagin Duality asserts that
the map
G W G ! OG; acting as G .x/./ D .x/ .x 2 G;2 OG/
implements a canonical algebraic and topological isomorphism between G and its
second dual OG.
Hom, Ext with Compact Second Argument In the sequel, the letters M; N will
stand for compact groups, so their duals OM; ON are discrete groups. The group
Hom. OM; N/ carries a compact topology: the compact-open topology makes it into a
compact group.
Lemma 5.1. For compact groups M and N, there are natural .topological/ isomorphisms
of compact groups
MN W Hom. OM; N/ ! Hom. ON; M/;
MN W Hom. OM; N/ ! Hom. OM ˝ ON; T/ D . OM ˝ ON/O: