##plugins.themes.bootstrap3.article.main##
Abstract
Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with the corresponding weak Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset.
It is proved that for $0<r<s\leq\infty$ the inclusion $\mathcal{B} \cap \mathcal{C}^{s}(M,N) \subset \mathcal{B}$ is a weak homotopy equivalence.
It is also established a parametrized variant of such a result.
In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $\mathcal{C}^{s}$ isotopies $\eta:[0,1]\times M \to M$ fixed near $\{0,1\}\times M$ into the space of loops $\Omega(\mathcal{D}^{r}(M), \mathrm{id}_{M})$ of the group of $\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\mathrm{id}_{M}$ is a weak homotopy equivalence.
Keywords:
weak homotopy equivalence, weak Whitney topology, smooth approximation
##plugins.themes.bootstrap3.article.details##
How to Cite
Khokhliuk, O., & Maksymenko, S. (2020). Smooth approximations and their applications to homotopy types. Proceedings of the International Geometry Center, 13(2), 68-108. https://doi.org/10.15673/tmgc.v13i2.1781
Issue
Section
Papers
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution (CC-BY) 4.0 License that allows others to share the work with an acknowledgment of the work’s authorship and initial publication in this journal.
Provided they are the owners of the copyright to their work, authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal’s published version of the work (e.g., post it to an institutional repository, in a journal or publish it in a book), with an acknowledgment of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories, disciplinary repositories, or on their website) prior to and during the submission process.
References
1. Hamza Alzaareer, Alexander Schmeding. Differentiable mappings on products with different degrees of differentiability in the two factors. Expo. Math., 33(2):184-222, 2015, doi: http://dx.doi.org/10.1016/j.exmath.2014.07.002.
2. Herbert Amann. Ordinary differential equations, volume 13 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1990, doi: http://dx.doi.org/10.1515/9783110853698. An introduction to nonlinear analysis, Translated from the German by Gerhard Metzen.
3. Habib Amiri, Helge Glockner, Alexander Schmeding. Lie groupoids of mappings taking values in a Lie groupoid. arXiv:1811.02888, 2018.
4. Bernd Ammann, Alexandru D. Ionescu, Victor Nistor. Sobolev spaces on Lie manifolds and regularity for polyhedral domains. Doc. Math., 11:161-206, 2006.
5. Jean Cerf. Topologie de certains espaces de plongements. Bull. Soc. Math. France, 89:227-380, 1961, http://www.numdam.org/item?id=BSMF_1961__89__227_0.
6. Adrian Clough. Reference request: Inclusion of smooth maps into continuous maps between smooth manifolds is a weak homotopy equivalence. 2016, https://math.stackexchange.com/questions/1794666.
7. Ralph L. Cohen, Andrew Stacey. Fourier decompositions of loop bundles. In Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory, volume 346 of Contemp. Math., pages 85-95. Amer. Math. Soc., Providence, RI, 2004, doi: http://dx.doi.org/10.1090/conm/346/06286.
8. Adrien Douady. Varietes `a bord anguleux et voisinages tubulaires. In Seminaire Henri Cartan, 1961/62, Exp. 1, page 11. Secretariat mathematique, Paris, 1961/1962, http://www.numdam.org/item/SHC_1961-1962__14__A1_0.
9. C. J. Earle, J. Eells. The diffeomorphism group of a compact Riemann surface. Bull. Amer. Math. Soc., 73:557-559, 1967, https://projecteuclid.org/euclid.bams/1183528956.
10. C. J. Earle, J. Eells. A fibre bundle description of Teichmuller theory. J. Differential Geometry, 3:19-43, 1969, doi: http://dx.doi.org/10.4310/jdg/1214428816.
11. José Figueroa-O'Farrill. Topology of function spaces? 2010, https://mathoverflow.net/questions/35180.
12. Ralph H. Fox. On topologies for function spaces. Bull. Amer. Math. Soc., 51:429-432, 1945, doi: http://dx.doi.org/10.1090/S0002-9904-1945-08370-0.
13. Helge Glockner. Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups. J. Funct. Anal., 194(2):347-409, 2002, doi: http://dx.doi.org/10.1006/jfan.2002.3942.
14. Helge Glockner. Homotopy groups of ascending unions of infinite-dimensional manifolds. arXiv:0812.4713, 2008.
15. C. Godbillon, G. Reeb. Fibres sur le branchement simple. Enseignement Math. (2), 12:277-287, 1966.
16. Marek Golasinski, Thiago de Melo, Edivaldo L. dos Santos. On path-components of the mapping spaces M(mathbbS^m,mathbbFP^n). Manuscripta Math., 158(3-4):401-419, 2019, doi: http://dx.doi.org/10.1007/s00229-018-1012-5.
17. M. Golubitsky, V. Guillemin. Stable mappings and their singularities. Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14.
18. Andre Gramain. Le type d'homotopie du groupe des diffeomorphismes d'une surface compacte. Ann. Sci. Ecole Norm. Sup. (4), 6:53-66, 1973, doi: http://dx.doi.org/10.24033/asens.1242.
19. David W. Henderson, James E. West. Triangulated infinite-dimensional manifolds. Bull. Amer. Math. Soc., 76:655-660, 1970, doi: http://dx.doi.org/10.1090/S0002-9904-1970-12478-8.
20. Morris W. Hirsch. Differential topology, volume 33 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original.
21. Sze-tsen Hu. Theory of retracts. Wayne State University Press, Detroit, 1965.
22. Andreas Kriegl, Peter W. Michor. Smooth and continuous homotopies into convenient manifolds agree. 2002, https://www.mat.univie.ac.at/ michor/homotopy.pdf.
23. S. Lojasiewicz. Sur le probl`eme de la division. Studia Math., 18:87-136, 1959, doi: http://dx.doi.org/10.4064/sm-18-1-87-136.
24. Jean-Pierre Magnot. Remarks on the geometry and the topology of the loop spaces H^s(S^1, N), for sleq 1/2. International Journal of Maps in Mathematics, hal-02285964:14-37, 2019.
25. Sergiy Maksymenko, Eugene Polulyakh. Foliations with non-compact leaves on surfaces. Proceedings of Geometric Center, 8(3-4):17-30, 2015 (in English).
26. Sergiy Maksymenko, Eugene Polulyakh. Foliations with all nonclosedleaves on noncompact surfaces. Methods Funct. Anal. Topology, 22(3):266-282, 2016.
27. Sergiy Maksymenko, Eugene Polulyakh. Actions of groups of foliated homeomorphisms on spaces of leaves. arxiv: 2006.01953, page 16, 2020 (in english).
28. B. Malgrange. Ideals of differentiable functions. Tata Institute of Fundamental Research Studies in Mathematics, No. 3. Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967.
29. Juan Margalef Roig, Enrique Outerelo Dominguez. Differential topology, volume 173 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1992. With a preface by Peter W. Michor.
30. Peter W. Michor. Manifolds of differentiable mappings, volume 3 of Shiva Mathematics Series. Shiva Publishing Ltd., Nantwich, 1980.
31. John Milnor. On spaces having the homotopy type of a rm CW-complex. Trans. Amer. Math. Soc., 90:272-280, 1959, doi: http://dx.doi.org/10.2307/1993204.
32. Amiya Mukherjee. Differential topology. Hindustan Book Agency, New Delhi; Birkhauser/Springer, Cham, second edition, 2015, doi: http://dx.doi.org/10.1007/978-3-319-19045-7.
33. Christoph Muller, Christoph Wockel. Equivalences of smooth and continuous principal bundles with infinite-dimensional structure group. Adv. Geom., 9(4):605-626, 2009, doi: http://dx.doi.org/10.1515/ADVGEOM.2009.032.
34. James R. Munkres. Topology. Prentice Hall, Inc., Upper Saddle River, NJ, 2000.
35. Karl-Hermann Neeb. Central extensions of infinite-dimensional Lie groups. Ann. Inst. Fourier (Grenoble), 52(5):1365-1442, 2002, doi: http://dx.doi.org/10.5802/aif.1921.
36. Richard S. Palais. Homotopy theory of infinite dimensional manifolds. Topology, 5:1-16, 1966, doi: http://dx.doi.org/10.1016/0040-9383(66)90002-4.
37. Valentin Poenaru. Un theor`eme des fonctions implicites pour les espaces d'applications C^infty . Inst. Hautes Etudes Sci. Publ. Math., (38):93-124, 1970, http://www.numdam.org/item?id=PMIHES_1970__38__93_0.
38. M. M. Postnikov. Lections in algebraic topology. \"Nauka\", Moscow, 1984. Elements of homotopy theory (in russian).
39. David Michael Roberts, Alexander Schmeding. Extending Whitney's extension theorem: nonlinear function spaces. https://arxiv.org/abs/1801.04126, 2018.
40. Katsuro Sakai. On topologies of triangulated infinite-dimensional manifolds. J. Math. Soc. Japan, 39(2):287-300, 1987, doi: http://dx.doi.org/10.2969/jmsj/03920287.
41. Stephen Smale. Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc., 10:621-626, 1959, doi: http://dx.doi.org/10.1090/S0002-9939-1959-0112149-8.
42. Samuel Bruce Smith. On the rational homotopy theory of function spaces. PhD thesis, 1993. Thesis (Ph.D.)-University of Minnesota.
43. Samuel Bruce Smith. The homotopy theory of function spaces: a survey. 519:3-39, 2010, doi: http://dx.doi.org/10.1090/conm/519/10228.
44. Andrew Stacey. Finite-dimensional subbundles of loop bundles. Pacific J. Math., 219(1):187-199, 2005, doi: http://dx.doi.org/10.2140/pjm.2005.219.187.
45. Andrew Stacey. Constructing smooth manifolds of loop spaces. Proc. Lond. Math. Soc. (3), 99(1):195-216, 2009, doi: http://dx.doi.org/10.1112/plms/pdn058.
46. Andrew Stacey. The smooth structure of the space of piecewise-smooth loops. Glasg. Math. J., 59(1):27-59, 2017, doi: http://dx.doi.org/10.1017/S0017089516000033.
47. Norman Steenrod. The topology of fibre bundles. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. Reprint of the 1957 edition, Princeton Paperbacks.
48. Robert M. Switzer. Algebraic topology - homotopy and homology. Classics in Mathematics. Springer-Verlag, Berlin, 2002. Reprint of the 1975 original [Springer, New York].
49. A. S. Svarc. On the homotopic topology of Banach spaces. Dokl. Akad. Nauk SSSR, 154:61-63, 1964.
50. Christoph Wockel. A generalization of Steenrod's approximation theorem. Arch. Math. (Brno), 45(2):95-104, 2009.
2. Herbert Amann. Ordinary differential equations, volume 13 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1990, doi: http://dx.doi.org/10.1515/9783110853698. An introduction to nonlinear analysis, Translated from the German by Gerhard Metzen.
3. Habib Amiri, Helge Glockner, Alexander Schmeding. Lie groupoids of mappings taking values in a Lie groupoid. arXiv:1811.02888, 2018.
4. Bernd Ammann, Alexandru D. Ionescu, Victor Nistor. Sobolev spaces on Lie manifolds and regularity for polyhedral domains. Doc. Math., 11:161-206, 2006.
5. Jean Cerf. Topologie de certains espaces de plongements. Bull. Soc. Math. France, 89:227-380, 1961, http://www.numdam.org/item?id=BSMF_1961__89__227_0.
6. Adrian Clough. Reference request: Inclusion of smooth maps into continuous maps between smooth manifolds is a weak homotopy equivalence. 2016, https://math.stackexchange.com/questions/1794666.
7. Ralph L. Cohen, Andrew Stacey. Fourier decompositions of loop bundles. In Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory, volume 346 of Contemp. Math., pages 85-95. Amer. Math. Soc., Providence, RI, 2004, doi: http://dx.doi.org/10.1090/conm/346/06286.
8. Adrien Douady. Varietes `a bord anguleux et voisinages tubulaires. In Seminaire Henri Cartan, 1961/62, Exp. 1, page 11. Secretariat mathematique, Paris, 1961/1962, http://www.numdam.org/item/SHC_1961-1962__14__A1_0.
9. C. J. Earle, J. Eells. The diffeomorphism group of a compact Riemann surface. Bull. Amer. Math. Soc., 73:557-559, 1967, https://projecteuclid.org/euclid.bams/1183528956.
10. C. J. Earle, J. Eells. A fibre bundle description of Teichmuller theory. J. Differential Geometry, 3:19-43, 1969, doi: http://dx.doi.org/10.4310/jdg/1214428816.
11. José Figueroa-O'Farrill. Topology of function spaces? 2010, https://mathoverflow.net/questions/35180.
12. Ralph H. Fox. On topologies for function spaces. Bull. Amer. Math. Soc., 51:429-432, 1945, doi: http://dx.doi.org/10.1090/S0002-9904-1945-08370-0.
13. Helge Glockner. Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups. J. Funct. Anal., 194(2):347-409, 2002, doi: http://dx.doi.org/10.1006/jfan.2002.3942.
14. Helge Glockner. Homotopy groups of ascending unions of infinite-dimensional manifolds. arXiv:0812.4713, 2008.
15. C. Godbillon, G. Reeb. Fibres sur le branchement simple. Enseignement Math. (2), 12:277-287, 1966.
16. Marek Golasinski, Thiago de Melo, Edivaldo L. dos Santos. On path-components of the mapping spaces M(mathbbS^m,mathbbFP^n). Manuscripta Math., 158(3-4):401-419, 2019, doi: http://dx.doi.org/10.1007/s00229-018-1012-5.
17. M. Golubitsky, V. Guillemin. Stable mappings and their singularities. Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14.
18. Andre Gramain. Le type d'homotopie du groupe des diffeomorphismes d'une surface compacte. Ann. Sci. Ecole Norm. Sup. (4), 6:53-66, 1973, doi: http://dx.doi.org/10.24033/asens.1242.
19. David W. Henderson, James E. West. Triangulated infinite-dimensional manifolds. Bull. Amer. Math. Soc., 76:655-660, 1970, doi: http://dx.doi.org/10.1090/S0002-9904-1970-12478-8.
20. Morris W. Hirsch. Differential topology, volume 33 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original.
21. Sze-tsen Hu. Theory of retracts. Wayne State University Press, Detroit, 1965.
22. Andreas Kriegl, Peter W. Michor. Smooth and continuous homotopies into convenient manifolds agree. 2002, https://www.mat.univie.ac.at/ michor/homotopy.pdf.
23. S. Lojasiewicz. Sur le probl`eme de la division. Studia Math., 18:87-136, 1959, doi: http://dx.doi.org/10.4064/sm-18-1-87-136.
24. Jean-Pierre Magnot. Remarks on the geometry and the topology of the loop spaces H^s(S^1, N), for sleq 1/2. International Journal of Maps in Mathematics, hal-02285964:14-37, 2019.
25. Sergiy Maksymenko, Eugene Polulyakh. Foliations with non-compact leaves on surfaces. Proceedings of Geometric Center, 8(3-4):17-30, 2015 (in English).
26. Sergiy Maksymenko, Eugene Polulyakh. Foliations with all nonclosedleaves on noncompact surfaces. Methods Funct. Anal. Topology, 22(3):266-282, 2016.
27. Sergiy Maksymenko, Eugene Polulyakh. Actions of groups of foliated homeomorphisms on spaces of leaves. arxiv: 2006.01953, page 16, 2020 (in english).
28. B. Malgrange. Ideals of differentiable functions. Tata Institute of Fundamental Research Studies in Mathematics, No. 3. Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967.
29. Juan Margalef Roig, Enrique Outerelo Dominguez. Differential topology, volume 173 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1992. With a preface by Peter W. Michor.
30. Peter W. Michor. Manifolds of differentiable mappings, volume 3 of Shiva Mathematics Series. Shiva Publishing Ltd., Nantwich, 1980.
31. John Milnor. On spaces having the homotopy type of a rm CW-complex. Trans. Amer. Math. Soc., 90:272-280, 1959, doi: http://dx.doi.org/10.2307/1993204.
32. Amiya Mukherjee. Differential topology. Hindustan Book Agency, New Delhi; Birkhauser/Springer, Cham, second edition, 2015, doi: http://dx.doi.org/10.1007/978-3-319-19045-7.
33. Christoph Muller, Christoph Wockel. Equivalences of smooth and continuous principal bundles with infinite-dimensional structure group. Adv. Geom., 9(4):605-626, 2009, doi: http://dx.doi.org/10.1515/ADVGEOM.2009.032.
34. James R. Munkres. Topology. Prentice Hall, Inc., Upper Saddle River, NJ, 2000.
35. Karl-Hermann Neeb. Central extensions of infinite-dimensional Lie groups. Ann. Inst. Fourier (Grenoble), 52(5):1365-1442, 2002, doi: http://dx.doi.org/10.5802/aif.1921.
36. Richard S. Palais. Homotopy theory of infinite dimensional manifolds. Topology, 5:1-16, 1966, doi: http://dx.doi.org/10.1016/0040-9383(66)90002-4.
37. Valentin Poenaru. Un theor`eme des fonctions implicites pour les espaces d'applications C^infty . Inst. Hautes Etudes Sci. Publ. Math., (38):93-124, 1970, http://www.numdam.org/item?id=PMIHES_1970__38__93_0.
38. M. M. Postnikov. Lections in algebraic topology. \"Nauka\", Moscow, 1984. Elements of homotopy theory (in russian).
39. David Michael Roberts, Alexander Schmeding. Extending Whitney's extension theorem: nonlinear function spaces. https://arxiv.org/abs/1801.04126, 2018.
40. Katsuro Sakai. On topologies of triangulated infinite-dimensional manifolds. J. Math. Soc. Japan, 39(2):287-300, 1987, doi: http://dx.doi.org/10.2969/jmsj/03920287.
41. Stephen Smale. Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc., 10:621-626, 1959, doi: http://dx.doi.org/10.1090/S0002-9939-1959-0112149-8.
42. Samuel Bruce Smith. On the rational homotopy theory of function spaces. PhD thesis, 1993. Thesis (Ph.D.)-University of Minnesota.
43. Samuel Bruce Smith. The homotopy theory of function spaces: a survey. 519:3-39, 2010, doi: http://dx.doi.org/10.1090/conm/519/10228.
44. Andrew Stacey. Finite-dimensional subbundles of loop bundles. Pacific J. Math., 219(1):187-199, 2005, doi: http://dx.doi.org/10.2140/pjm.2005.219.187.
45. Andrew Stacey. Constructing smooth manifolds of loop spaces. Proc. Lond. Math. Soc. (3), 99(1):195-216, 2009, doi: http://dx.doi.org/10.1112/plms/pdn058.
46. Andrew Stacey. The smooth structure of the space of piecewise-smooth loops. Glasg. Math. J., 59(1):27-59, 2017, doi: http://dx.doi.org/10.1017/S0017089516000033.
47. Norman Steenrod. The topology of fibre bundles. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. Reprint of the 1957 edition, Princeton Paperbacks.
48. Robert M. Switzer. Algebraic topology - homotopy and homology. Classics in Mathematics. Springer-Verlag, Berlin, 2002. Reprint of the 1975 original [Springer, New York].
49. A. S. Svarc. On the homotopic topology of Banach spaces. Dokl. Akad. Nauk SSSR, 154:61-63, 1964.
50. Christoph Wockel. A generalization of Steenrod's approximation theorem. Arch. Math. (Brno), 45(2):95-104, 2009.