Cohomology Theories for Compact Abelian Groups

1. Front Matter

   Pages 1-12

   PDF

2. Introduction

   • Karl H. Hofmann, Paul S. Mostert

   Pages 13-22

3. Algebraic background

   • Karl H. Hofmann, Paul S. Mostert

   Pages 23-97

4. The cohomology of finite abelian groups

   • Karl H. Hofmann, Paul S. Mostert

   Pages 98-154

5. The cohomology of classifying spaces of compact groups

   • Karl H. Hofmann, Paul S. Mostert

   Pages 155-171

6. Kan extensions of functors on dense categories

   • Karl H. Hofmann, Paul S. Mostert

   Pages 172-202

7. The cohomological structure of compact abelian groups

   • Karl H. Hofmann, Paul S. Mostert

   Pages 203-223

8. Appendix

   • Karl H. Hofmann, Paul S. Mostert

   Pages 224-228

9. Back Matter

   Pages 229-236

   PDF

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.

• Abelian group
• Algebraic structure
• Cohomology
• Group theory
• Representation theory

• Dept. of Mathematics, Tulane University, New Orleans, USA

  Karl H. Hofmann, Paul S. Mostert

Policies and ethics