  We consider the near-horizon geometries of extremal, rotating black hole
solutions of the vacuum Einstein equations, including a negative cosmological
constant, in four and five dimensions. We assume the existence of one
rotational symmetry in 4d, two commuting rotational symmetries in 5d and in
both cases non-toroidal horizon topology. In 4d we determine the most general
near-horizon geometry of such a black hole, and prove it is the same as the
near-horizon limit of the extremal Kerr-AdS(4) black hole. In 5d, without a
cosmological constant, we determine all possible near-horizon geometries of
such black holes. We prove that the only possibilities are one family with a
topologically S^1 X S^2 horizon and two distinct families with topologically
S^3 horizons. The S^1 X S^2 family contains the near-horizon limit of the
boosted extremal Kerr string and the extremal vacuum black ring. The first
topologically spherical case is identical to the near-horizon limit of two
different black hole solutions: the extremal Myers-Perry black hole and the
slowly rotating extremal Kaluza-Klein (KK) black hole. The second topologically
spherical case contains the near-horizon limit of the fast rotating extremal KK
black hole. Finally, in 5d with a negative cosmological constant, we reduce the
problem to solving a sixth-order non-linear ODE of one function. This allows us
to recover the near-horizon limit of the known, topologically S^3, extremal
rotating AdS(5) black hole. Further, we construct an approximate solution
corresponding to the near-horizon geometry of a small, extremal AdS(5) black
ring.
