A short proof that \(\mathrm{Diff}_{c}(M)\) is perfect (Q258906)

This paper presents a short proof that for a smooth manifold \(M\) (excluding \(\mathbb R\) in this proof) the group Diff\(_c(M)\) of diffeomorphisms of \(M\) which are isotopic to the identity through a compactly supported isotopy is perfect. In fact, given \(g\in \text{Diff}_c(M)\) there are time 1 maps \(f_i\) of a vector field \(X_i\) (\(i=1,\dots,r\)) on \(M\) so that \(g=[g_1,f_1]\dots[g_r,f_r]\).