Local triviality of the restriction map for immersions with normal crossings: An extension of a result of R. Palais (Q1122835)

The author proves the following result: Let B be a compact smooth manifold with boundary and closed submanifold M in the interior of B, Y a smooth manifold without boundary and dim Y\(=\dim B\). Let r: Imm(B,Y)\(\to Imm(M,Y)\) be the restriction map for the spaces of smooth immersions, \(Imm_ x(M,Y)\) be immersions with normal crossings and \(S=r^{- 1}(Imm_ x(M,Y))\). Then \(\hat r=r| S\) is a locally trivial fibration. The author points out that there are examples where r itself is not a fibration for dim B\(=\dim Y\), while r is known to be a (not necessarily locally trivial) fibration for dim B\(<\dim Y\). the problem is of interest in continuum mechanics. For the very clear proof the author constructs a group homomorphism s: \(G\to {\mathfrak G}\) together with natural actions of \({\mathfrak G}\) resp. G on Imm(M,Y) resp. Imm(B,Y), which restrict to \(Imm_ x(M,Y)\) resp. S. A local section of s about \(1_ G\) and the compatibility of the actions with r and s then is used to explicitly construct the locally trivial structure for \(\hat r.\) A very important point, of course, is the stability of normal crossings.