New proofs of theorems of Kathryn Mann

Abstract

We give a shorter proof of the following theorem of Kathryn Mann [M]: the identity component of the group of the compactly supported C^r diffeomorphisms of Rⁿ cannot admit a nontrivial C^p-action on S¹, provided n ≥ 2, r ≠ n + 1 and p ≥ 2. We also give a new proof of another theorem of Mann [M]: any nontrivial homomorphism from the group of the orientation preserving C^r diffeomorphisms of the circle to the group of C^p diffeomorphisms of the circle is the conjugation of the standard inclusion by a C^p diffeomorphism, if r ≥ p, r ≠ 2 and p ≠ 1.