Composite differentiable functions (Q5917817)

In the early 1940s, Whitney proved that every \({\mathcal C}^\infty\) even function \(f(x)\) can be written \(f(x)= g(x^2)\), where \(g\) is \({\mathcal C}^\infty\) [\textit{H. Whitney}, Duke Math. J. 10, 159-160 (1943; Zbl 0063.08235)]. About twenty years later, Glaeser (answering a question posed by Thom in connection with the \({\mathcal C}^\infty\) preparation theorem) showed that a \({\mathcal C}^\infty\) function \(f(x)= f(x_1, \dots, x_m)\) which is invariant under permutations of the coordinates can be expressed \(f(x)= g(\sigma_1(x), \dots, \sigma_m(x))\), where \(g\) is \({\mathcal C}^\infty\) and the \(\sigma_i(x)\) are the elementary symmetric polynomials [\textit{G. Glaeser}, Ann. Math. II. Ser. 77, 193-209 (1963; Zbl 0106.31302)]. Of course, not every \({\mathcal C}^\infty\) function \(f(x)\) which is constant on the fibres of a (proper or semiproper) real analytic mapping \(y=\varphi(x)\), \(y=(y_1, \dots, y_n)\), can be expressed as a composite \(f=g \circ \varphi\), where \(g\) is \({\mathcal C}^\infty\). The real analytic mapping \(\varphi\) is said to have the \({\mathcal C}^\infty\) composite function property if every \({\mathcal C}^\infty\) function \(f(x)\) which is ``formally a composite with \(\varphi\)'' can be written \(f=g \circ \varphi\), where \(g(y)\) is \({\mathcal C}^\infty\). The theorem of Glaeser asserts that a semiproper real analytic mapping \(\varphi\) which is generally a submersion has the \({\mathcal C}^\infty\) composite function property. In this article the authors introduce a new point of view towards Glaeser's theorem, with respect to which it is formulated a ``\({\mathcal C}^k\) composite function property'' that is satisfied by all semiproper real analytic mappings. As a consequence, a closed subanalytic set \(X\) satisfies the \({\mathcal C}^\infty\) composite function property iff the ring \({\mathcal C}^\infty (X)\) of \({\mathcal C}^\infty \) functions on \(X\) is the intersection of all finite differentiability classes.