  We describe a method to find the anomaly of the time-reversal symmetry of
2+1d topological quantum field theories, by computing the fractional anomalous
momentum on the cross-cap background. This allows us, for example, to identify
the parameter $\nu$ mod 16 of the bulk 3+1d topological superconductor with
$\mathsf{T}^2=(-1)^F$ on whose boundary a given 2+1d time-reversal-invariant
topological phase can appear.
