A new topology for the set \(\text{C}^{\infty,G}(M,N)\) of \(G\)-equivariant smooth maps
Suppose \(M\) and \(N\) are \(C^r\) manifolds, \(1 \leq r \leq \omega\). Let \(C^r(M,N)\) denote the set of \(C^r\) maps from \(M\) to \(N\). The standard choices for a topology on \(C^r(M,N)\) are the weak topology, and the strong topology (also called the Whitney topology). In the paper under review, the authors give a new topology on \(C^r(M,N)\), which is called the strong-weak topology, and is at least as fine as the weak topology and at most as fine as the strong topology. If \(G\) is a Lie group and \(M\), \(N\) are \(C^r\) \(G\)-manifolds, \(C^{r,G} (M,N)\) denotes the set of \(G\)-equivariant \(C^r\) maps from \(M\) to \(N\). The authors also define the strong-weak topology on \(C^{r,G} (M,N)\) to be the relative topology of that of \(C^r (M,N)\). The authors show basic properties of the strong-weak topology. Among others, they show that the strong-weak topology agrees with the strong topology when \(G\) is compact, and that the strong-weak topology agrees with the weak topology when the orbit space \(M/G\) is compact.
scientific article; zbMATH DE number 1406574