A generalization of Steenrod’s approximation theorem

Christoph Wockel

Archivum Mathematicum (2009)

  • Volume: 045, Issue: 2, page 95-104
  • ISSN: 0044-8753

Abstract

top
In this paper we aim for a generalization of the Steenrod Approximation Theorem from [16, Section 6.7], concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalization is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a continuous section in a smooth locally trivial bundles can always be smoothed out in a very controlled way (in terms of the graph topology on spaces of continuous functions), preserving the section on regions where it is already smooth.

How to cite

top

Wockel, Christoph. "A generalization of Steenrod’s approximation theorem." Archivum Mathematicum 045.2 (2009): 95-104. <http://eudml.org/doc/250554>.

@article{Wockel2009,
abstract = {In this paper we aim for a generalization of the Steenrod Approximation Theorem from [16, Section 6.7], concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalization is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a continuous section in a smooth locally trivial bundles can always be smoothed out in a very controlled way (in terms of the graph topology on spaces of continuous functions), preserving the section on regions where it is already smooth.},
author = {Wockel, Christoph},
journal = {Archivum Mathematicum},
keywords = {infinite-dimensional manifold; infinite-dimensional smooth bundle; smoothing of continuous sections; density of smooth in continuous sections; topology on spaces of continuous functions; infinite-dimensional manifold; infinite-dimensional smooth bundle; smoothing; continuous sections; density; space of continuous functions},
language = {eng},
number = {2},
pages = {95-104},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A generalization of Steenrod’s approximation theorem},
url = {http://eudml.org/doc/250554},
volume = {045},
year = {2009},
}

TY - JOUR
AU - Wockel, Christoph
TI - A generalization of Steenrod’s approximation theorem
JO - Archivum Mathematicum
PY - 2009
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 045
IS - 2
SP - 95
EP - 104
AB - In this paper we aim for a generalization of the Steenrod Approximation Theorem from [16, Section 6.7], concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalization is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a continuous section in a smooth locally trivial bundles can always be smoothed out in a very controlled way (in terms of the graph topology on spaces of continuous functions), preserving the section on regions where it is already smooth.
LA - eng
KW - infinite-dimensional manifold; infinite-dimensional smooth bundle; smoothing of continuous sections; density of smooth in continuous sections; topology on spaces of continuous functions; infinite-dimensional manifold; infinite-dimensional smooth bundle; smoothing; continuous sections; density; space of continuous functions
UR - http://eudml.org/doc/250554
ER -

References

top
  1. Bourbaki, N., General topology, Elements of Mathematics (Berlin) (1998), Springer-Verlag, Berlin, translated from the French. (1998) Zbl0894.54001
  2. Dugundji, J., Topology, Allyn and Bacon Inc., Boston, 1966. (1966) Zbl0144.21501MR0193606
  3. Glöckner, H., Infinite-dimensional Lie groups without completeness restrictions, Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups (Bȩdlewo, 2000), vol. 55, Banach Center Publ., 43–59, Polish Acad. Sci., Warsaw, 2002. (2002) Zbl1020.58009MR1911979
  4. Glöckner, H., Neeb, K.-H., Infinite-dimensional Lie groups, Basic Theory and Main Examples, volume I, Springer-Verlag, 2009, in preparation. (2009) 
  5. Hamilton, R. S., 10.1090/S0273-0979-1982-15004-2, Bull. Amer. Math. Soc. (N.S.) 7 (1) (1982), 65–222. (1982) Zbl0499.58003MR0656198DOI10.1090/S0273-0979-1982-15004-2
  6. Hirsch, M. W., Differential Topology, Springer-Verlag, New York, 1976. (1976) Zbl0356.57001MR0448362
  7. Keller, H. H., Differential calculus in locally convex spaces, Lecture Notes in Math., vol. 417, Springer-Verlag, Berlin, 1974. (1974) Zbl0293.58001MR0440592
  8. Kriegl, A., Michor, P. W., Smooth and continuous homotopies into convenient manifolds agree, unpublished preprint, 2002, available from http://www.mat.univie.ac.at/michor/. 
  9. Kriegl, A., Michor, P. W., The Convenient Setting of Global Analysis, Math. Surveys Monogr., vol. 53, Amer. Math. Soc., 1997. (1997) Zbl0889.58001MR1471480
  10. Lee, J. M., 10.1007/978-0-387-21752-9, Grad. Texts in Math., vol. 218, Springer-Verlag, New York, 2003. (2003) MR1930091DOI10.1007/978-0-387-21752-9
  11. Michor, P. W., Manifolds of Differentiable Mappings, Shiva Mathematics Series, vol. 3, Shiva Publishing Ltd., Nantwich, 1980, out of print, online available from http://www.mat.univie.ac.at/michor/. (1980) Zbl0433.58001MR0583436
  12. Milnor, J., Remarks on infinite-dimensional Lie groups, Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984, pp. 1007–1057. (1984) Zbl0594.22009MR0830252
  13. Müller, C., Wockel, C., Equivalences of smooth and continuous principal bundles with infinite-dimensional structure group, Adv. Geom., to appear, 2009, arXiv:math/0604142. 
  14. Naimpally, S. A., 10.1090/S0002-9947-1966-0192466-4, Trans. Amer. Math. Soc. 123 (1966), 267–272. (1966) Zbl0151.29703MR0192466DOI10.1090/S0002-9947-1966-0192466-4
  15. Neeb, K.-H., 10.5802/aif.1921, Ann. Inst. Fourier (Grenoble) 52 (5) (2002), 1365–1442. (2002) Zbl1019.22012MR1935553DOI10.5802/aif.1921
  16. Steenrod, N., The Topology of Fibre Bundles, Princeton Math. Ser. 14 (1951). (1951) Zbl0054.07103MR0039258
  17. Wockel, C., Smooth extensions and spaces of smooth and holomorphic mappings, J. Geom. Symmetry Phys. 5 (2006), 118–126. (2006) Zbl1108.58006MR2269885

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.