We prove that Alexander-Spanier cohomology with coefficients in a topologicalAbelian group G is isomorphic to the group of isomorphism classes of principal bundles with certain Abelian structure groups. The result holds if either X is a CW-space and G arbitrary or if X is metrizable or compact Hausdorff and G an ANR.
The non-commutative torus is realized as the -algebra of sections of a locally trivial -algebra bundle over with fibres isomorphic to for a totally skew multiplier on . D. Poguntke [9] proved that is stably isomorphic to for a simple non-commutative torus and an integer . It is well-known that a stable isomorphism of two separable -algebras is equivalent to the existence of equivalence bimodule between them. We construct an --equivalence bimodule.
We show that for n ≥ 3 the symplectic group Sp(n) is as a 2-compact group determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows us to determine the integral homotopy type of Sp(n) among connected finite loop spaces with maximal torus.