  The topology of the domain of outer communication for 5-dimensional
stationary bi-axisymmetric black holes is classified in terms of disc bundles
over the 2-sphere and plumbing constructions. In particular we find an
algorithmic bijective correspondence between the plumbing of disc bundles and
the rod structure formalism for such spacetimes. Furthermore, we describe a
canonical fill-in for the black hole region and cap for the asymptotic region.
The resulting compactified domain of outer communication is then shown to be
homeomorphic to $S^4$, a connected sum of $S^2\times S^2$'s, or a connected sum
of complex projective planes $\mathbb{CP}^2$. Combined with recent existence
results, it is shown that all such topological types are realized by vacuum
solutions. In addition, our methods treat all possible types of asymptotic
ends, including spacetimes which are asymptotically flat, asymptotically
Kaluza-Klein, or asymptotically locally Euclidean.
