Diffeomorphisms lying in one-parameter groups and extension of stratified homeomorphisms (Q2710563)

The author proves a conjecture of Milnor from 1984: For a \(C^r\)-manifold \(S\) diffeomorphic to the interior of a compact manifold, the connected component \(\text{Diff}^r_0(S,S)\) of the identity map \(1_S: S \to S\) in the group of all \(C^r\)-diffeomorphisms of \(S\) is the image of the exponential map \(E\) defined on the space of \(C^r\)-vector fields on \(S\) admitting a global flow. In other words, every diffeomorphism in \(\text{Diff}^r_0(S,S)\) can be written as a composition of diffeomorphisms defined by the flows at time \(t = 1\) of finitely many vector fields on \(S\). The proof uses techniques and results developed by McDuff. For \(S\) a closed manifold, it is known that \(\text{Diff}^r_0(S,S)\) is a simple group, which allows one to prove the result about the image of the exponential map corresponding to the aforesaid. For \(S\) a non-compact manifold, the simplicity of \(\text{Diff}^r_0(S,S)\) may fail to be true, setting the conjecture of Milnor in perspective. The author uses his result to obtain two extension theorems for stratified maps defined on some strata of a stratified space. The paper is very carefully written.