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.

I. Algebraic background.- Section 1. On exponential functors.- Definition 1.1. Multiplicative category, exponential functor and polynomial algebra, Hopf algebra - Definition 1.2. Subadditive and sub-multiplicative functors, compatible natural transformations - Lemma 1.3. E2SA and TE1A are algebras when Ei is exponential, S subadditive, T submultiplicative - Lemma 1.4. ? is a morphism of algebras - Lemma 1.5. About coalgebras - Proposition 1.6. E HomR (-, M) ? HomZ (E -, M) is a morphism of graded algebras for E = ?, P - Lemmas 1.10, 1.11. The structure of Hom(PZn, Z) - Lemmas 1.12, 1.13. More about Hom (E-, M) - Proposition 1.14. The structure of Hom (EA, M), E = ?, P - Definition 1.15. Polynomial algebras with divided powers - Proposition 1.16. E Hom (-, M) ? Hom (E -, M) for E = P ? ? - Lemma 1.17. About the natural map ?A Proposition 1.18. The coalgebra Hom (G, Z) - Corollary 1.19. The duality of polynomial algebras and algebras with divided powers - Theorem 1.22. The map PRA ? R ? RB ? Hom ($$\hat PA \otimes {\text{ }} \wedge {\text{ }}\hat B,R$$) - Proposition 1.23. about HomS (K ? L, A) ? HomS (L, Hom (K, A)) for complexes.- Section 2. The arithmetic of certain spectral algebras.- Definition 2.1. Spectral algebra, edge algebra - Lemmas 2.2, 2.3. The derivations d, d?, - Definition 2.4. The functors E2, E3 - Lemma 2.5. The cohomology map preserves multiplication - Lemma 2.6. Definition of the cohomology map ? - Definition 2.7. The first edge algebra and B2P (?) - Definition 2.8. Integral elements in rings, weakly principal ideal rings - Definition 2.10. The formalism of the derivation d? on E2(?) - Definition 2.11. The elementary morphisms - Proposition 2.12. The structure of the edge terms in E3(?) - Lemma 2.13. The elements of ker d?, - Lemma 2.14. The elements of im d? - Proposition 2.16. u ? a8' ? ua8': E3II(?) ? a8' ? E3(?) is injective - Proposition 2.17. The terms next to the edge terms - An explicit example - Corollary 2.18. The terms next to the edge terms for a principal ideal domain as coefficient ring - Lemma 2.20. Passage to the ring of quotients in the coefficient ring - Proposition 2.21. E2(? ? ?) ? E2(?) ? E2(?) - Proposition 2.24, 2.25. Conditions under which d? is exact - Proposition 2.26. The exactness of d? within the ground ring extension - Lemma 2.31. Elementary morphisms yielding the same E3 - Proposition 2.32. The case that ? is a homothety - Proposition 2.33. Elementary morphisms which differ by a scalar - Proposition 2.34. E3(?1 ? ?2) ? E3(?2) if im d(?1) is flat - Proposition 2.35. An inductive process to compute E3(?) if the ground ring is a principal ideal domain - Theorem I. E3(?) is generated as a (P coker ?)-module by M - Definition 2.39. Definition of ? and E2*(?) - Lemma 2.40. The differential modules (E2(?), d') - Proposition 2.42. About the structure of E3(?) - Propositions 2.43, 2.44. About the PA-module structure of Er(?) - Propositions 2.47, 2.48. Non-injective elementary morphisms.- Section 3. Some analogues of the results about spectral algebras with dual derivations.- Lemma 3.1. The differential and derivative ?? - Definition 3.2, 3.3. The spectral algebras Er[?], Er{?} - Lemma 3.4. E2 [-] is an exponential functor - Lemma 3.5. About ?d + d? - Proposition 3.6. The edge algebra E3II [?] - Definition 3.6a. R-coalgebras, differential graded co-algebras, differential graded Hopf algebras - Proposition 3.7. E2{?} is a differential bi-graded Hopf algebra relative to d?, and ?? - Lemma 3.8. The cofunctor f ? E2 {Hom8 (f, R)} - Lemma 3.9. About the structure of finite abelian groups - Definition 3.10. Standard resolution of a finite abelian group - Lemma 3.11. The uniqueness of standard resolutions -Lemma 3.12. The four term exact sequence derived from an injection - Lemma 3.13. Isomorphic version of ker ?pHom (f, A) - Proposition 3.14. The edge terms in E3 (Hom (f, R)) - Corollary 3.15. The morphism PR Ext (G, R) ?R Hom (? G, R) ? E3 (Hom (f, R)) - Corollary 3.16. The functoriality of this morphism - Propositions 3.17, 3.18. The isomorphisms H (R/Z ? E2(f)) ? E3(f) ? H (E2(f)?)?..- Section 4. The Bockstein formalism.- Lemmas 4.1, 4.2, 4.3, 4.4. Some diagram chasing - Definition 4.5. The definition of pre-Bockstein diagrams and standard Bockstein diagrams - Lemmas 4.6, 4.7, 4.8. About the Bockstein formalism - Proposition 4.9. An isomorphism of exact sequences - Lemma 4.10. More diagram chasing - Proposition 4.11. Sufficient conditions for the Bockstein formalism for complexes - Proposition 4.12. When is the Bockstein differential a derivation? - Corollaries 4.13, 4.14. The standard situation - Proposition 4.15. The Bockstein formalism for the cohomology of groups and complexes - Proposition 4.16. The Bockstein formalism for the spectral algebras E2(?) of Section 2 - Corollary 4.17. A particular case of 4.16..- II. The cohomology of finite abelian groups.- Section 1. Products.- Definition 1.1. The construction of ? - Definition 1.2. The construction of ? - Lemma 1.3. Tensoring resolutions - Corollary 1.4 - Lemma 1.5. The Künneth theorem - Theorem 1.6. The resolution of augmented Hopf algebras - Theorem II. Cohomology and the tensor product of Hopf algebras - Corollary 1.7. About H(G1 × G2, R) - Corollary 1.8. A Künneth theorem for H(G1 × G2, R) - Corollary 1.9. A special case of 1.8 - Corollary 1.10. H(G1 × G2, R) for cyclic G1 - Corollary 1.11. H(G1, R) ? ? ? H(Gn, R) ? H(G1 × ? × Gn, R) - Corollary 1.12. About the annihilator of H+(G1 × G2, R) - Corollary 1.13. About the exponent of H+(G1 × G2, R) - Corollary 1.14. The exponent of H+(G, M) for a finite abelian group G and arbitrary M - Corollary 1.15. H(G, M) ? N ? H(G, M ? N).- Section 2. Special free resolutions for finite abelian groups.- Definition 2.1. Special elements in the group ring of a finite abelian group - Lemma 2.2. About ?: ? ? ? A+ - Lemma 2.3. d? ?d = 0 - Lemma 2.4. The coderivation D = d + ? - Definition 2.5. E(f) and Ê (f) - Lemma 2.6. Ê is exponential - Lemma 2.7. Ê(f) exact - special case - Lemma 2.8. 0 ? Z ? Ê (f) is a resolution - Lemma 2.9. R ?SA Horns (HomS (A, S), R) - Theorem III. Fundamental theorem about the cohomology of finite abelian groups - Lemma 2.10. Hi (G, R/Z) ? Hi+1(G, Z) - Proposition 2.11. Various isomorphisms involving H(G, R/Z) - Lemma 2.12. A categorical lemma - Theorem 2.13. The morphism ?: PR Ext(G, R) ?R Hom(?…