  Using methods from algebraic topology and group cohomology, I pursue
Grothendieck's question on equality of geometric and cohomological Brauer
groups in the context of complex-analytic spaces. The main result is that
equality holds under suitable assumptions on the fundamental group and the
Pontrjagin dual of the second homotopy group. I apply this to Lie groups, Hopf
manifolds, and complex-analytic surfaces.
