Abstract
This chapter contains foundational material on spaces of diffeomorphisms and embeddings. Such spaces are known to be Fréchet manifolds, separable when the manifolds involved are compact. Versions of these and related facts are developed for manifolds with boundary, as well as in the context of fiber-preserving diffeomorphisms and maps. The latter utilizes a modification of the exponential map, called the aligned exponential, adapted to the fibered structure.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aneziris, C., Balachandran, A.P., Bourdeau, M., Jo, S., Sorkin, R.D., Ramadas, T.R.: Aspects of spin and statistics in generally covariant theories. Int. J. Mod. Phys. A 4(20), 5459–5510 (1989)
Asano, K.: Homeomorphisms of prism manifolds. Yokohama Math. J. 26(1), 19–25 (1978)
Banyaga, A.: The structure of classical diffeomorphism groups. In: Mathematics and Its Applications, vol. 400. Kluwer Academic, Dordrecht (1997)
Bessaga, C., Pelczynski, A.: Selected topics in infinite-dimensional topology. In: Monografie Matematyczne, Tom 58 [Mathematical Monographs, vol. 58]. PWN—Polish Scientific Publishers, Warsaw (1975)
Boileau, M., Otal, J.P.: Scindements de Heegaard et groupe des homeotopies des petites varietes de Seifert. Invent. Math. 106(1), 85–107 (1991)
Bonahon, F.: Difféotopies des espaces lenticulaires. Topology 22(3), 305–314 (1983)
Bredon, G.E., Wood, J.W.: Non-orientable surfaces in orientable 3-manifolds. Invent. Math. 7, 83–110 (1969)
Bruce, J.W.: On transversality. Proc. Edinb. Math. Soc. (2) 29(1), 115–123 (1986)
Casson, A.J., Gordon, C.McA.: Reducing Heegaard splittings. Topology Appl. 27(3), 275–283 (1987)
Cerf, J.: Topologie de certains espaces de plongements. Bull. Soc. Math. Fr. 89, 227–380 (1961)
Cerf, J.: Sur les difféomorphismes de la sphere de dimension trois \(({\Gamma }_{4} = 0)\). In: Lecture Notes in Mathematics, vol. 53. Springer, Berlin (1968)
Charlap, L.S., Vasquez, A.T.: Compact flat riemannian manifolds. III. The group of affinities. Am. J. Math. 95, 471–494 (1973)
Filipkiewicz, R.P.: Isomorphisms between diffeomorphism groups. Ergod. Theor. Dyn. Syst. 2(2), 159–171 (1982)
Friedman, J., Sorkin, R.: Spin 1/2 from gravity. Phys. Rev. Lett. 44, 1100–1103 (1980)
Gabai, D.: The Smale conjecture for hyperbolic 3-manifolds: \(\mathrm{Isom}(\mathrm{{M}}^{3}) \simeq \mathrm{ Diff}(\mathrm{{M}}^{3})\). J. Differ. Geom. 58(1), 113–149 (2001)
Gibson, C.G., Wirthmuller, K., du Plessis, A.A., Looijenga, E.J.N.: Topological stability of smooth mappings. In: Lecture Notes in Mathematics, vol. 552. Springer, Berlin (1976)
Giulini, D.: On the configuration space topology in general relativity. Helv. Phys. Acta 68(1), 86–111 (1995)
Giulini, D., Louko, J.: No-boundary θ sectors in spatially flat quantum cosmology. Phys. Rev. D (3) 46(10), 4355–4364 (1992)
Gramain, A.: Le type d’homotopie du groupe des difféomorphismes d’une surface compacte. Ann. Sci. École Norm. Sup. (4) 6, 53–66 (1973)
Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. (N.S.) 7(1), 65–222 (1982)
Hantsche, W., Wendt, W.: Drei dimensionali Euklidische Raumformen. Math. Ann. 110, 593–611 (1934)
Hatcher, A.E.: Homeomorphisms of sufficiently large P 2-irreducible 3-manifolds. Topology 15(4), 343–347 (1976)
Hatcher, A.E.: On the diffeomorphism group of \({S}^{1} \times {S}^{2}\). Proc. Am. Math. Soc. 83(2), 427–430 (1981)
Hatcher, A.E.: A proof of the Smale conjecture, \(\mathrm{Diff}({S}^{3}) \simeq \mathrm{ O}(4)\). Ann. Math. (2) 117(3), 553–607 (1983)
Hatcher, A.: On the diffeomorphism group of \({S}^{1} \times {S}^{2}\). Revised version posted at http://www.math.cornell.edu/~hatcher/#papers (2003)
Hempel, J.: 3-Manifolds. In: Annals of Mathematics Studies, vol. 86. Princeton University Press, Princeton (1976)
Henderson, D.W.: Corrections and extensions of two papers about infinite-dimensional manifolds. Gen. Topology Appl. 1, 321–327 (1971)
Henderson, D.W., Schori, R.: Topological classification of infinite dimensional manifolds by homotopy type. Bull. Am. Math. Soc. 76, 121–124 (1970)
Hendriks, H.: La stratification naturelle de l’espace des fonctions différentiables reelles n’est pas la bonne. C. R. Acad. Sci. Paris Ser. A-B 274, A618–A620 (1972)
Isham, C.J.: Topological θ-sectors in canonically quantized gravity. Phys. Lett. B 106(3), 188–192 (1981)
Ivanov, N.V.: Groups of diffeomorphisms of Waldhausen manifolds, Studies in topology, II. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 66, 172–176, 209 (1976)
Ivanov, N.V.: Diffeomorphism groups of Waldhausen manifolds. J. Sov. Math. 12, 115–118 (1979)
Ivanov, N.V.: Homotopies of automorphism spaces of some three-dimensional manifolds. Dokl. Akad. Nauk SSSR 244(2), 274–277 (1979)
Ivanov, N.V.: Corrections: Homotopies of automorphism spaces of some three-dimensional manifolds. Dokl. Akad. Nauk SSSR 249(6), 1288 (1979)
Ivanov, N.V.: Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 122, 72–103, 164–165 (1982)
Ivanov, N.V.: Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds. J. Soviet Math. 26, 1646–1664 (1984)
Jaco, W.: Lectures on three-manifold topology. In: CBMS Regional Conference Series in Mathematics, vol. 43. American Mathematical Society, Providence (1980)
Jaco, W., Shalen, P.B.: Seifert fibered spaces in 3-manifolds. Memoir. Am. Math. Soc. 21(220), viii+192 (1979)
Kalliongis, J., McCullough, D.: Isotopies of 3-manifolds. Topology Appl. 71(3), 227–263 (1996)
Karcher, H.: Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30(5), 509–541 (1977)
Kobayashi, T., Saeki, O.: The Rubinstein-Scharlemann graphic of a 3-manifold as the discriminant set of a stable map. Pac. J. Math. 195(1), 101–156 (2000)
Kriegl, A., Michor, P.W.: The convenient setting of global analysis. In: Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997)
Friedman, J.L., Witt, D.M.: Homotopy is not isotopy for homeomorphisms of 3-manifolds. Topology 25(1), 35–44 (1986)
Lundell, A., Weingram, S.: The Topology of CW Complexes. Van Nostrand Reinhold, Princeton (1969)
Mather, J.N.: Stability of \(\mathrm{{C}}^{\infty }\) mappings. III. Finitely determined mapgerms. Inst. Hautes Etudes Sci. Publ. Math. 35, 279–308 (1968)
McCullough, D.: Isometries of elliptic 3-manifolds. J. Lond. Math. Soc. (2) 65(1), 167–182 (2002)
McCullough, D., Soma, T.: The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold (2010) [ArXiv:1005.5061]
Neumann, W., Raymond, F.: Automorphisms of Seifert manifolds (1979). Preprint
Orlik, P.: Seifert manifolds. In: Lecture Notes in Mathematics, vol. 291. Springer, Berlin (1972)
Orlik, P., Vogt, E., Zieschang, H.: Zur Topologie gefaserter dreidimensionaler Mannigfaltigkeiten. Topology 6, 49–64 (1967)
Palais, R.S.: Local triviality of the restriction map for embeddings. Comment. Math. Helv. 34, 305–312 (1960)
Palais, R.S.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966)
Park, C.Y.: Homotopy groups of automorphism groups of some Seifert fiber spaces. Dissertation at the University of Michigan (1989)
Park, C.Y.: On the weak automorphism group of a principal bundle, product case. Kyungpook Math. J. 31(1), 25–34 (1991)
Pitts, J.T., Rubinstein, J.H.: Applications of minimax to minimal surfaces and the topology of 3-manifolds. Miniconference on geometry and partial differential equations, 2 (Canberra, 1986), pp. 137–170. In: Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 12. Austral. Nat. Univ., Canberra (1987)
Rubinstein, J.H.: On 3-manifolds that have finite fundamental group and contain Klein bottles. Trans. Am. Math. Soc. 251, 129–137 (1979)
Rubinstein, J.H., Birman, J.S.: One-sided Heegaard splittings and homeotopy groups of some 3-manifolds. Proc. Lond. Math. Soc. (3) 49(3), 517–536 (1984)
Rubinstein, J.H., Scharlemann, M.: Comparing Heegaard splittings of non-Haken 3-manifolds. Topology 35(4), 1005–1026 (1996)
Sakuma, M.: The geometries of spherical Montesinos links. Kobe J. Math. 7(2), 167–190 (1990)
Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983)
Seeley, R.T.: Extension of C∞ functions defined in a half space. Proc. Am. Math. Soc. 15, 625–626 (1964)
Seifert, H.: Topologie dreidimensionaler gefaserter raume. Acta Math. 60(1), 147–238 (1933)
Sergeraert, F.: Un théoréme de fonctions implicites sur certains espaces de Fréchet et quelques applications. Ann. Sci. École Norm. Sup. (4) 5, 599–660 (1972)
Smale, S.: Diffeomorphisms of the 2-sphere. Proc. Am. Math. Soc. 10, 621–626 (1959)
Sorkin, R.D.: Classical topology and quantum phases: Quantum geons. In: Geometrical and Algebraic Aspects of Nonlinear Field Theory. North-Holland Delta Series, pp. 201–218. North-Holland, Amsterdam (1989)
Takens, F.: Characterization of a differentiable structure by its group of diffeomorphisms. Bol. Soc. Brasil. Mat. 10(1), 17–25 (1979)
Tougeron, J.C.: Une généralisation du théoréme des fonctions implicites. C. R. Acad. Sci. Paris Ser. A-B 262, A487–A489 (1966)
Tougeron, J.C.: Idéaux de fonctions différentiables. I. Ann. Inst. Fourier (Grenoble) 18(fasc. 1), 177–240 (1968)
Waldhausen, F.: Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II. (German). Invent. Math. 3, 308–333 (1967). Ibid: 4, 87–117 (1967)
Waldhausen, F.: Gruppen mit Zentrum und 3-dimensionale Mannigfaltigkeiten. Topology 6, 505–517 (1967)
Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. (2) 87, 56–88 (1968)
Wall, C.T.C.: Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13(6), 481–539 (1981)
Witt, D.M.: Symmetry groups of state vectors in canonical quantum gravity. J. Math. Phys. 27(2), 573–592 (1986)
Wolf, J.A.: Spaces of Constant Curvature, 3rd edn. Publish or Perish Inc., Boston (1974)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hong, S., Kalliongis, J., McCullough, D., Rubinstein, J.H. (2012). Diffeomorphisms and Embeddings of Manifolds. In: Diffeomorphisms of Elliptic 3-Manifolds. Lecture Notes in Mathematics, vol 2055. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31564-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-31564-0_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31563-3
Online ISBN: 978-3-642-31564-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)