  Symmetry-protected topological (SPT) phases of matter have been interpreted
in terms of anomalies, and it has been expected that a similar picture should
hold for SPT phases with fermions. Here, we describe in detail what this
picture means for phases of quantum matter that can be understood via band
theory and free fermions. The main examples we consider are time-reversal
invariant topological insulators and superconductors in 2 or 3 space
dimensions. Along the way, we clarify the precise meaning of the statement that
in the bulk of a 3d topological insulator, the electromagnetic $\theta$-angle
is equal to $\pi$.
