  On top of the mean-field analysis of a Bose-Einstein condensate, one
typically applies the Bogoliubov theory to analyze quantum fluctuations of the
excited modes. Therefore, one has to diagonalize the Bogoliubov Hamiltonian in
a symplectic manner. In our article we investigate the topology of these
Bogoliubov excitations in inversion-invariant systems of interacting bosons. We
analyze how the condensate influences the topology of the Bogoliubov
excitations. Analogously to the fermionic case, here we establish a symplectic
extension of the polarization characterizing the topology of the Bogoliubov
excitations and link it to the eigenvalues of the inversion operator at the
inversion-invariant momenta. We also demonstrate an instructive but
experimentally feasible example that this quantity is also related to edge
states in the excitation spectrum.
