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Full Description
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.
Contents
I. Algebraic background.- Section 1. On exponential functors.- Section 2. The arithmetic of certain spectral algebras.- Section 3. Some analogues of the results about spectral algebras with dual derivations.- Section 4. The Bockstein formalism.- II. The cohomology of finite abelian groups.- Section 1. Products.- Section 2. Special free resolutions for finite abelian groups.- Section 3. About the cohomology of finite abelian groups in the case of trivial action.- Section 4. Appendix to Section 3: The low dimensions.- III. The cohomology of classifying spaces of compact groups.- Section 1. The functor h.- Section 2. The functor h for finite groups.- IV. Kan extensions of functors on dense categories.- Section 1. Dense categories and continuous functors.- Section 2. Multiplicative Hopf extensions.- V. The cohomological structure of compact abelian groups.- Section 1. The cohomologies of connected compact abelian groups.- Section 2. The space cohomology of arbitrary compact abelian groups.- Section 3. The canonical embedding of ? in hG.- Section 4. Cohomology theories for compact groups over fields as coefficient domains.- Section 5. The structure of h for arbitrary compact abelian groups and integral coefficients.- VI. Appendix. Another construction of the functor h.- Proposition 1. About the graph of < for a topological monoid acting on a space — Proposition 2. Properties of the Dold-Lashof spectrum.- List of notations.
