A note about \(G\)-equivariant homotopies and isotopies, where \(G\) is a Lie group
Let \(G\) be a good Lie group, i.e., a closed subgroup of a Lie group having only finitely many connected components. For example, every linear Lie group is good. Let \(M\) be a proper \(C^r\) \(G\)-manifold \((1 \leq r \leq \omega)\), and \(N\) a \(C^r\) \(G\)-manifold. Assume that \(N\) is also proper if \(G\) has infinitely many connected components. The author proves that (1) a continuous \(G\)-map \(M \to N\) is \(G\)-homotopic to a \(C^r\) \(G\)-map by a homotopy which approximates the constant homotopy, (2) two \(G\)-homotopic \(C^r\) \(G\)-maps \(M \to N\) are \(C^r\) \(G\)-homotopic. He also proves that if \(M\) is a proper \(C^{\omega}\) \(G\)-manifold, then (1) a \(C^{\infty}\) \(G\)-diffeomorphism \(M \to M\) is \(C^{\infty}\) \(G\)-isotopic to a \(C^{\omega}\) \(G\)-diffeomorphism, and (2) two \(C^{\infty}\) \(G\)-isotopic \(C^{\omega}\) \(G\)-diffeomorphisms \(M \to M\) are \(C^{\omega}\) \(G\)-isotopic. The proofs of the case where \(G\) has only finitely many connected components are based on the existence of global slices of proper \(G\)-manifolds.
scientific article; zbMATH DE number 1346259