Question

Cylinder homotopy

Answer 1

If fₜ were such a homotopy then you'd get a map H:S^(1)×I → S^(1) given by H(θ,t) = pr(fₜ(0,θ/2ϖ)), where pr is projection onto the first factor. The twice-stationary hypothesis ensures that H is well-defined, namely that H(0,t) = H(2ϖ,t) for all t. But H(θ,0) = pr(f(0,θ/2ϖ)) = θ and H(θ,1) = pr(id(0,θ/2ϖ)) = 0, so H provides a homotopy between the degree 1 map θ↦θ of S^(1) and the degree 0 (constant) map θ↦0, a contradiction.