Exercise 2 solution: Let k be an integer and x a real number. πR_{ω}(x) = x + ω + k mod 1 = x + ω mod 1. On the other hand, r_{ω}π(x) = (x mod 1 + ω) mod 1 = x + ω mod 1. Thus R_{ω} is a lift of r_{ω}.