We prove a uniqueness theorem for stationary $D$-dimensional Kaluza-Klein black holes with $D-2$ Killing fields, generating the symmetry group ${ \mathbb R} \times U(1)^{ D-3}$. It is shown that the topology and metric of such black holes is uniquely determined by the angular momenta and certain other invariants consisting of a number of real moduli, as well as integer vectors subject to certain constraints.
Categories: gr-qc