  We discuss the topology of Bogoliubov excitation bands from a Bose-Einstein
condensate in an optical lattice. Since the Bogoliubov equation for a bosonic
system is non-Hermitian, complex eigenvalues often appear and induce dynamical
instability. As a function of momentum, the onset of appearance and
disappearance of complex eigenvalues is an exceptional point (EP), which is a
point where the Hamiltonian is not diagonalizable and hence the Berry
connection and curvature are ill-defined, preventing defining topological
invariants. In this paper, we propose a systematic procedure to remove EPs from
the Brillouin zone by introducing an imaginary part of the momentum. We then
define the Berry phase for a one-dimensional bosonic Bogoliubov system.
Extending the argument for Hermitian systems, the Berry phase for an
inversion-symmetric system is shown to be $Z_2$. As concrete examples, we
numerically investigate two toy models and confirm the bulk-edge correspondence
even in the presence of complex eigenvalues. The $Z_2$ invariant associated
with particle-hole symmetry and the winding number for a
time-reversal-symmetric system are also discussed.
