implies that the exterior region is topologically ${\Bbb R} \times V^{(4)}$ with $V^{(4)}$ being some four-dimensional simply connected Riemannian manifold. However, the simple connectedness by itself does not completely determine the topology of $V^{(4)}$. Therefore, in the present theorem, we simply demand that $V^{(4)} \approx \{{\Bbb R}^4 \setminus {\Bbb B}^4 \}$, which is in accordance with the topology of the Cveti[č]{}-Youm solution with equal charges. Our boundary conditions—in particular, the [*rod structure*]{}, which was first introduced by Harmark [@Harmark] based on earlier work for static solutions [@weyl]—are accordingly specified in the manner discussed in Sec. \[sec:coincidence\]. The topology theorem [@Cai; @Helfgott; @galloway] yields that in five-dimensions, cross-sections of the event horizon must be topologically either a sphere, a ring, or a lens-space. The requirement (2) excludes some interesting class of solutions to be dealt with. It would be interesting to consider generalization of our uniqueness theorem to include solutions with non-spherical horizon topology. We would like to emphasize that even under these restrictive assumptions (1) and (2), still it is not at all obvious whether black holes in EMCS-theory are uniquely specified by their global charges. In fact, it has been shown by numerical studies [@Kunz] that when the value 