Higher singularities of smooth functions are unnecessary (Q799277)

Let \({\mathcal E}_ n\) be the ring of germs at 0 of smooth functions f: \({\mathbb{R}}^ n\to {\mathbb{R}}\). Let \({\mathcal M}_ n\subset {\mathcal E}_ n\) be the unique maximal ideal consisting of germs of functions f where \(f(0)=0\). If \(f\in {\mathcal M}^ 2_ n\) then the Jacobian ideal of f is the ideal J(f) in \({\mathcal E}_ n\) which is generated by \(\partial f/\partial x_ i (i=1,...,n)\). The codimension of f is defined by \(\dim_{{\mathbb{R}}}{\mathcal M}_ n/J(f)\). If f: \({\mathbb{R}}^ n\to {\mathbb{R}}\) is a smooth function and \(x\in {\mathbb{R}}\), let \(j_ x(f)\) be the germ at 0 of the function \(g(y)=f(x+y)\). Let \(j^ s_ x(f)\in J^ s(n)={\mathcal E}_ n/{\mathcal M}_ n^{s+1}\) be the reduction modulo \({\mathcal M}_ n^{s+1}\) of \(j_ x(f)\). Let \(\tilde j^ s_ x(f)=j^ s_ x(f)-f(x)\in\tilde J^ s(n)={\mathcal M}_ n/{\mathcal M}_ n^{s+1}.\) We call \(\tilde j^ s_ x(f)\) the normalized s jet of f at x. Let N be a compact n-manifold (with boundary). Let f: \({\mathbb{N}}\to {\mathbb{R}}\) be a smooth function and \(x\in N\). We say that f has a higher singularity at x if codimension of f at x is at least 2. The main theorem in this paper is stated as follows. Let g: \(N\to [0,1]\) be a smooth map which is non-singular near \(\partial N\) and define \({\mathcal H}(N,g)\) to be the space of all smooth maps f: \(N\to [0,1]\) such that \(f=g\) near \(\partial N\) and f has no ''higher singularities''. Let \(\tilde J^ s(N)\to N\) be the bundle of normalized s-jets of maps \(N\to {\mathbb{R}}\). Any smooth map f: \(N\to {\mathbb{R}}\) gives a section \(\tilde j^ sf\) of \(J^ s(N)\). Let \({\mathcal H}^ s\) be the subbundle of \(\tilde J^ s(N)\) consisting of normalized s-jets of maps f: \(N\to {\mathbb{R}}\) with no higher singularities. Let \(\Gamma^ s_{{\mathcal H}}\) be the space of all sections \(\sigma\) : \(N\to {\mathcal H}^ s\) such that \(\sigma =\tilde j^ sg\) near \(\partial N\). If \(f\in {\mathcal H}(N,g)\) then \(\tilde j^ sf\in\Gamma^ s_{{\mathcal H}}.\) Theorem: \(\tilde j^ s: {\mathcal H}(N,g)\to\Gamma^ s_{{\mathcal H}}\) is n- connected if \(s\geq 3\). The main application of this theorem is the following. Let M be a compact smooth (n-1)-manifold, let \(N=M\times [0,1]\) and g: \(N\to [0,1]\) be the projection map. Let \({\mathcal C}(M)=Diff(M\times [0,1];rel M\times 0\cup\partial M\times [0,1])\) with Whitney topology. Then \({\mathcal C}(M)\) is the space of pseudoisotopies of M. By the result of Cerf, \({\mathcal C}(M)\simeq {\mathcal E}(M)=the\) space of smooth maps f: \(N\to [0,1]\) without critical points such that \(f=g\) near \(\partial N\). Using the above theorem the author shows that the map \(\pi_ k({\mathcal H}(N,g),{\mathcal E}(M))\to\pi_{k-1}({\mathcal E}(M))\) is a naturally split epimorphism if \(k<n.\) At any rate this work is very beautiful and I think that it is one of the important application of Mather's local theory of singularities to the global theory of pseudoisotopy.