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We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding \mathbb {N}\trianglelefteq \mathbb {E} of locally compact groups and a twisted action (\alpha,\tau) thereof on a (post)liminal C^*-algebra A the twisted crossed product A\rtimes _{\alpha,\tau }\mathbb {E} is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup \mathbb {N}\trianglelefteq \mathbb {E} is type-I as soon as \mathbb {E} is. This happens for instance if \mathbb {N} is discrete and \mathbb {E} is Lie, or if \mathbb {N} is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group \mathbb {G} type-I-preserving if all semidirect products \mathbb {N}\rtimes \mathbb {G} are type-I as soon as \mathbb {N} is, and linearly type-I-preserving if the same conclusion holds for semidirect products V\rtimes \mathbb {G} arising from finite-dimensional \mathbb {G}-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie. [ABSTRACT FROM AUTHOR]

Let X be a Hopf manifold with non-Abelian fundamental group and E be a holomorphic vector bundle over X, with trivial pull-back to ℂn − {0}. The authors show that there exists a line bundle L over X such that E ⊗ L has a nowhere vanishing section. It is proved that in case dim(X) ≥ 3, π*(E) is trivial if and only if E is filtrable by vector bundles. With the structure theorem, the authors get the cohomology dimension of holomorphic bundle E over X with trivial pull-back and the vanishing of Chern class of E. [ABSTRACT FROM AUTHOR]