  For a black hole's spacetime manifold in the Euclidean signature, its metric
is positive definite and therefore a Riemannian manifold. It can be regarded as
a gravitational instanton and a topological characteristic which is the Euler
number is associated. In this paper we derive a formula for the Euler numbers
of four-dimensional rotating black holes by the integral of the Euler density
on the spacetime manifolds of black holes. Using this formula, we obtain that
the Euler numbers of Kerr and Kerr-Newman black holes are 2. We also obtain
that the Euler number of the Kerr-Sen metric in the heterotic string theory
with one boost angle nonzero is 2 that is in accordence with its topology.
