The Method of Cerf and Palais

Sungbok Hong, John Kalliongis, Darryl McCullough, J. Hyam Rubinstein

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

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 we will prove the

Original languageEnglish
Title of host publicationDiffeomorphisms of Elliptic 3-Manifolds
PublisherSpringer Verlag
Pages19-51
Number of pages33
ISBN (Print)9783642315633
DOIs
Publication statusPublished - 2012 Jan 1

Publication series

NameLecture Notes in Mathematics
Volume2055
ISSN (Print)0075-8434
ISSN (Electronic)1617-9692

Keywords

  • Exceptional Orbit
  • Klein Bottle
  • Lens Space
  • Seifert Manifold
  • Singular Fiberings

ASJC Scopus subject areas

  • Algebra and Number Theory

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  • Cite this

    Hong, S., Kalliongis, J., McCullough, D., & Rubinstein, J. H. (2012). The Method of Cerf and Palais. In Diffeomorphisms of Elliptic 3-Manifolds (pp. 19-51). (Lecture Notes in Mathematics; Vol. 2055). Springer Verlag. https://doi.org/10.1007/978-3-642-31564-0_3