Of all topological algebraic structures compact topological groups
have perhaps the richest theory since 80 many different fields
contribute to their study: Analysis enters through the
representation theory and harmonic analysis; differential geo
metry, the theory of real analytic functions and the theory of
differential equations come into the play via Lie group theory;
point set topology is used in describing the local geometric
structure of compact groups via limit spaces; global topology and
the theory of manifolds again playa role through Lie group theory;
and, of course, algebra enters through the cohomology and homology
theory. A particularly well understood subclass of compact groups
is the class of com pact abelian groups. An added element of
elegance is the duality theory, which states that the category of
compact abelian groups is completely equivalent to the category of
(discrete) abelian groups with all arrows reversed. This allows for
a virtually complete algebraisation of any question concerning
compact abelian groups. The subclass of compact abelian groups is
not so special within the category of compact. groups as it may
seem at first glance. As is very well known, the local geometric
structure of a compact group may be extremely complicated, but all
local complication happens to be "abelian." Indeed, via the duality
theory, the complication in compact connected groups is faithfully
reflected in the theory of torsion free discrete abelian groups
whose notorious complexity has resisted all efforts of complete
classification in ranks greater than two."
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