The Method of Cerf and Palais

A fundamental theorem of R. Palais and J. Cerf shows that the map sending a diffeomorphism to its restriction to an imbedding of a submanifold is a locally trivial fibration of the spaces of mappings involved. In this chapter, this theorem is extended to other maps between spaces of mappings, in particular to the map sending each fiber-preserving diffeomorphism of a bundle to the diffeomorphism it induces on the base manifold. Versions with boundary control are obtained, as well as versions for singular fiberings, a class that includes Seifert-fibered three-manifolds. A final section gives a proof that sending the space of diffeomorphisms of a singularly fibered manifold to its space of cosets by the subgroup of fiber-preserving diffeomorphisms is a fibration. This coset space may be regarded as the space of fibered structures diffeomorphic to the given one.R. Palais (Comment. Math. Helv. 34:305–312, 1960) proved a very useful result relating diffeomorphisms and embeddings. For closed M, it says that if are submanifolds of M, then the mappings and obtained by restricting diffeomorphisms and embeddings are locally trivial, and hence are Serre fibrations. The same results, with variants for manifolds with boundary and more complicated additional boundary structure, were proven by J. Cerf (Bull. Soc. Math. Fr. 89:227–380, 1961). Among various applications of these results, the Isotopy Extension Theorem follows by lifting a path in starting at the inclusion map of V to a path in. Moreover, parameterized versions of isotopy extension follow just as easily from the homotopy lifting property for (see Corollary 3.3).In this chapter, we will extend the theorem of Palais in various ways. Many of our results concern fiber-preserving maps. For example, in Sect. 3.3