  We investigate topological properties and classification of mean-field
theories of stable bosonic systems. Of the three standard classifying
symmetries, only time-reversal represents a real symmetry of the many-boson
system, while the other two, particle-hole and chiral, are simply constraints
that manifest as symmetries of the effective single-particle problem. For
gapped systems in arbitrary space dimension we establish three fundamental
no-go theorems that prove the absence of: parity switches,
symmetry-protected-topological quantum phases, and localized bosonic zero modes
under open boundary conditions. We then introduce a squaring, kernel-preserving
map connecting non-interacting Hermitian theories of fermions and stable boson
systems, which serves as a playground to reveal the role of topology in bosonic
phases and their localized midgap boundary modes. Finally, we determine the
symmetry classes inherited from the fermionic tenfold-way classification,
unveiling an elegant threefold-way topological classification of
non-interacting bosons. We illustrate our main findings in one- and
two-dimensional bosonic lattice and field-theory models.
