On time-reversal anomaly of 2+1d topological phases

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 ν mod 16 of the bulk 3+1d topological superconductor with T²=(-1)F on whose boundary a given 2+1d time-reversal-invariant topological phase can appear.