Diffeomorphism groups of balls and spheres (Q374088)

Denote by Diff\(^r_0(M)\) the group of isotopically trivial C\(^r\)-diffeomorphisms of the manifold \(M\). If \(M\) has a boundary then the homomorphism Diff\(^r_0(M)\to\) Diff\(^r_0(\partial M)\) induced by restriction is surjective. By way of contrast it is shown that there is no non-trivial homomorphism Diff\(^\infty_0(\mathbb S^{2k-1})\to\) Diff\(^1_0(\mathbb B^m)\) for any \(k,m\geq 1\). This is also in contrast to the situation for the corresponding groups of isotopically trivial homeomorphisms. It is also shown that there is a finitely-generated, torsion-free group \(\Gamma\) and a homomorphism \(\Gamma\to\) Diff\(^\infty(\mathbb S^1)\) which does not extend to a C\(^1\) action of \(\Gamma\) on \(\mathbb B^2\).