Squaring the fermion: The threefold way and the fate of zero modes

In the spirit of Dirac's derivation of a fermionic theory by "taking the square root" of the bosonic Klein-Gordon equation, we present a squaring procedure, mapping fermionic to bosonic theories, that helps establish a threefold way topological classification of stable noninteracting bosonic matter. The ephemeral nature of boundary states and the topological triviality of noninteracting, zero-temperature bosonic phases are supported by three no-go theorems, although excitations of those same phases may display topologically nontrivial characteristics.