  We investigate the topological classification of excitations in quadratic
bosonic systems with an excitation band gap. Time-reversal, charge-conjugation,
and parity symmetries in bosonic systems are introduced to realize a ten-fold
symmetry classification. We find a specific decomposition of the quadratic
bosonic Hamiltonian and use it to prove that each quadratic bosonic system is
homotopic to a direct sum of two single-particle subsystems. The topological
classification table is thus derived via inheriting from that of
Atland-Zirnbauer classes and unique topological phases of bosons are predicted.
Finally, concrete topological models are proposed to demonstrate the
peculiarity of bosonic excitations.
