The very-strong C\(^\infty\) topology on C\(^{\infty}(M,N)\) and \(K\)-equivariant maps
A main result of the paper is Theorem A asserting that the set \(C^{\omega}(M,N)\) of real analytic maps between real analytic manifolds \(M,N\) is dense in the space \(C^{\infty}_{vS}(M,N)\) of all infinitely differentiable maps between \(M\) and \(N\), endowed with so-called very strong \(C^\infty\)-topology. This theorem also has an equivariant counterpart proved in Theorem C. Namely, if the manifolds \(M,N\) are endowed with smooth actions of a compact Lie group \(K\), then the space \(C^{\omega,K}(M,N)\) of real analytic \(K\)-equivariant maps from \(M\) to \(N\) is dense in the subspace \(C^{\infty,K}_{vS}(M,N)\) of \(C^\infty_{vS}(M,N)\) consisting of \(K\)-equivariant smooth maps, provided the number of \(K\)-isotropy types in \(N\) is finite. (The latter condition allows to find an equivariant embedding of \(N\) into a Euclidean \(K\)-space and to reduce the problem to maps into a suitable \({\mathbb R}^m\)). A substantial part of the paper is devoted to discussing a true topology on the space \(C^r(M,N)\) of \(C^r\)-maps from \(M\) to \(N\), where \(1\leq r\leq\infty\). In case of finite \(r\) such a true topology on \(C^r(M,N)\) is the Whitney topology. For infinite \(r\) there are two natural choices for a true topology on \(C^\infty(M,N)\): the \textit{strong topology}, which is the projective limit of the induced Whitney \(C^r\)-topologies with \(r<\infty\); and the \textit{very strong topology} which is the strongest topology making the restriction operators \(C^\infty(M,N)\to C^\infty(B,N)\) open and continuous for any compact subset \(B\subset M\), where \(C^\infty(B,N)\) is endowed with the strong topology. Thus two functions \(f,g:M\to N\) are close in the very strong topology if they have near derivatives \(D^\alpha f(x)\), \(D^\alpha g(x)\) up to some order \(| \alpha| \leq r\) growing as \(x\in M\) tends to infinity. The author proves many basic properties of the very strong topology, in particular, Glue Lemma B asserting that for any \(f\in C^\infty(M,N)\) and any open set \(U\subset M\), the restriction \(f| U\) has a neighborhood \(\mathcal U\) in \(C^\infty(U,N)\) such that for any function \(g\in\mathcal U\) the composed map \(E(g):M\to N\) defined by \(E(g)| U=g\) and \(E(g)| M\setminus U=f| M\setminus U\) is infinitely smooth. Moreover the extension operator \(E:C^\infty(U,N)\to C^\infty(M,N)\), \(E:g\mapsto E(g)\), is continuous. This favorably distinguishes the very strong topology from the strong topology failing to have this glue property. These nice properties make the very strong topology a better choice for a true topology on \(C^\infty(M,N)\). Because of that the author proposes to change the existing terminology and refer to the very strong topology as the Whitney topology and to the strong topology as the Mather topology (in honor of \textit{J. Mather} who introduced the strong topology on \(C^\infty(M,N)\) in [Ann. Math. 89, 254--291 (1969; Zbl 0177.26002)] referring to it as the Whitney topology).
scientific article; zbMATH DE number 2002265