  Short-range entangled topological phases of matter are closely connected to
Topological Quantum Field Theory. We use this connection to classify bosonic
Symmetry Protected Topological Phases in low dimensions, including the case
when the symmetry involves time-reversal. To accomplish this, we generalize
Turaev's description of equivariant TQFT to the unoriented case. We show that
invertible unoriented equivariant TQFTs in one or less spatial dimensions are
classified by twisted group cohomology, in agreement with the group cohomology
proposal of Chen, Gu, Liu and Wen. We also show that invertible oriented
equivariant TQFTs in spatial dimension two or less are classified by ordinary
group cohomology.
