  We discuss a way to construct a commuting projector Hamiltonian model for a
(3+1)d topological superconductor in class DIII. The wave function is given by
a sort of string net of the Kitaev wire, decorated on the time reversal (T)
domain wall. Our Hamiltonian is provided on a generic 3d manifold equipped with
a discrete form of the spin structure. We will see how the 3d spin structure
induces a 2d spin structure (called a Kasteleyn direction on a 2d lattice) on T
domain walls, which makes possible to define fluctuating Kitaev wires on them.
Upon breaking the T symmetry in our model, we find the unbroken remnant of the
symmetry which is defined on the time reversal domain wall. The domain wall
supports the 2d non-trivial SPT protected by the unbroken symmetry, which
allows us to determine the SPT classification of our model, based on the recent
QFT argument by Hason, Komargodski, and Thorngren.
