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 language | English |
|---|---|
| Title of host publication | Diffeomorphisms of Elliptic 3-Manifolds |
| Publisher | Springer Verlag |
| Pages | 19-51 |
| Number of pages | 33 |
| ISBN (Print) | 9783642315633 |
| DOIs | |
| Publication status | Published - 2012 Jan 1 |
Publication series
| Name | Lecture Notes in Mathematics |
|---|---|
| Volume | 2055 |
| ISSN (Print) | 0075-8434 |
| ISSN (Electronic) | 1617-9692 |
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Keywords
- Exceptional Orbit
- Klein Bottle
- Lens Space
- Seifert Manifold
- Singular Fiberings
ASJC Scopus subject areas
- Algebra and Number Theory
Cite this
The Method of Cerf and Palais. / Hong, Sungbok; Kalliongis, John; McCullough, Darryl; Rubinstein, J. Hyam.
Diffeomorphisms of Elliptic 3-Manifolds. Springer Verlag, 2012. p. 19-51 (Lecture Notes in Mathematics; Vol. 2055).Research output: Chapter in Book/Report/Conference proceeding › Chapter
}
TY - CHAP
T1 - The Method of Cerf and Palais
AU - Hong, Sungbok
AU - Kalliongis, John
AU - McCullough, Darryl
AU - Rubinstein, J. Hyam
PY - 2012/1/1
Y1 - 2012/1/1
N2 - 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
AB - 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
KW - Exceptional Orbit
KW - Klein Bottle
KW - Lens Space
KW - Seifert Manifold
KW - Singular Fiberings
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U2 - 10.1007/978-3-642-31564-0_3
DO - 10.1007/978-3-642-31564-0_3
M3 - Chapter
AN - SCOPUS:85072883779
SN - 9783642315633
T3 - Lecture Notes in Mathematics
SP - 19
EP - 51
BT - Diffeomorphisms of Elliptic 3-Manifolds
PB - Springer Verlag
ER -