  Extending recent work in 5 dimensions, we prove the existence and uniqueness
of solutions to the reduced Einstein equations for vacuum black holes in
$(n+3)$-dimensional spacetimes admitting the isometry group $\mathbb{R}\times
U(1)^{n}$, with Kaluza-Klein asymptotics for $n\geq3$. This is equivalent to
establishing existence and uniqueness for singular harmonic maps $\varphi:
\mathbb{R}^3\setminus\Gamma\rightarrow SL(n+1,\mathbb{R})/SO(n+1)$ with
prescribed blow-up along $\Gamma$, a subset of the $z$-axis in $\mathbb{R}^3$.
We also analyze the topology of the domain of outer communication for these
spacetimes, by developing an appropriate generalization of the plumbing
construction used in the lower dimensional case. Furthermore, we provide a
counterexample to a conjecture of Hollands-Ishibashi concerning the topological
classification of the domain of outer communication. A refined version of the
conjecture is then presented and established in spacetime dimensions less than
8.
