Fermionic symmetry protected topological phases and cobordisms

It has been proposed recently that interacting Symmetry Protected Topological Phases can be classified using cobordism theory. We test this proposal in the case of Fermionic SPT phases with $$ {\mathrm{\mathbb{Z}}}_2 $$ symmetry, where $$ {\mathrm{\mathbb{Z}}}_2 $$ is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known Fermionic SPT phases in space dimension D ≤ 3 and also predicts that all such phases can be realized by free fermions. In higher dimensions we predict the existence of inherently interacting fermionic SPT phases.