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Homeomorphism groups of commutator width one


Author: Takashi Tsuboi
Journal: Proc. Amer. Math. Soc. 141 (2013), 1839-1847
MSC (2010): Primary 54H15, 54H20, 57S05; Secondary 20F65, 37B05, 57N50
DOI: https://doi.org/10.1090/S0002-9939-2012-11595-3
Published electronically: November 28, 2012
MathSciNet review: 3020870
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Abstract: We show that any element of the identity component $ \mathrm {Homeo}(S^n)_0$ of the group of homeomorphisms of the $ n$-dimensional sphere $ S^n$ can be written as one commutator. We also show that any element of the group $ \mathrm {Homeo}(\mu ^n)$ of homeomorphisms of the $ n$-dimensional Menger compact space $ \mu ^n$ can be written as one commutator.


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  • 1. R. D. Anderson, The algebraic simplicity of certain groups of homeomorphisms, Amer. J. Math. 80 (1958), 955-963. MR 0098145 (20:4607)
  • 2. A. Banyaga, The structure of classical diffeomorphism groups, Mathematics and its Applications, vol. 400, Kluwer, Dordrecht, 1997. MR 1445290 (98h:22024)
  • 3. M. Bestvina, Characterizing k-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 71 (1988). MR 920964 (89g:54083)
  • 4. M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76. MR 0117695 (22:8470b)
  • 5. M. Brown, Locally flat imbeddings of topological manifolds, Ann. of Math. (2) 75 (1962), 331-341. MR 0133812 (24:A3637)
  • 6. M. Brown and H. Gluck, Stable structures on manifolds. I. Homeomorphisms of $ S^n$, Ann. of Math. (2) 79 (1) (1964), 1-17. MR 0158383 (28:1608a)
  • 7. D. Burago, S. Ivanov and L. Polterovich, Conjugation-invariant norms on groups of geometric origin, Adv. Stud. Pure Math., 52, Groups of Diffeomorphisms, Math. Soc. Japan, 2008, pp. 221-250. MR 2509711 (2011c:20074)
  • 8. A. Chigogidze, Compact spaces lying in the $ n$-dimensional universal Menger compact space and having homeomorphic complement in it, Math. USSR-Sb. 61 (1988), 471-484. MR 911804 (89c:54071)
  • 9. A. Chigogidze, K. Kawamura and E. D. Tymchatyn, Menger manifolds, Continua, with Houston problem book, Marcel Dekker, New York, 1995, pp. 37-88. MR 1326834 (96a:57052)
  • 10. D. B. A. Epstein, The simplicity of certain groups of homeomorphisms, Compositio Math. 22 (1970), 165-173. MR 0267589 (42:2491)
  • 11. A. Fathi, Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 1, 45-93. MR 584082 (81k:58042)
  • 12. G. M. Fisher, On the group of all homeomorphisms of a manifold, Trans. Amer. Math. Soc. 97 (1960), 193-212. MR 0117712 (22:8487)
  • 13. M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5-234. MR 538680 (81h:58039)
  • 14. R. Kirby, Stable homeomorphisms and the annulus conjecture, Ann. of Math. (2) 89 (3) (1969), 575-582. MR 0242165 (39:3499)
  • 15. J. Mather, The vanishing of the homology of certain groups of homeomorphisms, Topology 10 (1971), 297-298. MR 0288777 (44:5973)
  • 16. J. Mather, Commutators of diffeomorphisms. I, II and III, Comment. Math. Helv. 49 (1974), 512-528; 50 (1975), 33-40; and 60 (1985), 122-124. MR 0356129 (50:8600); MR 0375382 (51:11576); MR 0787665 (86g:58025)
  • 17. F. Quinn, Ends of maps. III. Dimensions $ 4$ and $ 5$, J. Differential Geometry 17 (1982), 503-521. MR 679069 (84j:57012)
  • 18. V. Sergiescu and T. Tsuboi, Acyclicity of the groups of homeomorphisms of the Menger compact spaces, Amer. J. Math. 118 (1996), 1299-1312. MR 1420925 (97g:54053)
  • 19. W. Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 304-307. MR 0339267 (49:4027)
  • 20. T. Tsuboi, On the uniform perfectness of diffeomorphism groups, Groups of Diffeomorphisms, Adv. Stud. Pure Math., 52, Math. Soc. Japan, 2008, pp. 505-524. MR 2509724 (2010k:57063)
  • 21. T. Tsuboi, On the uniform simplicity of diffeomorphism groups, Differential Geometry, World Sci. Publ., Hackensack, NJ, 2009, pp. 43-55. MR 2523489 (2010k:57062)
  • 22. T. Tsuboi, On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds, Comment. Math. Helv. 87 (2012), 141-185. MR 2874899
  • 23. S. M. Ulam and J. von Neumann, On the group of homeomorphisms of the surface of the sphere (Abstract), Bull. Amer. Math. Soc. 53 (1947), 508.

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Additional Information

Takashi Tsuboi
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Komaba Meguro, Tokyo 153-8914, Japan
Email: tsuboi@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2012-11595-3
Keywords: Homeomorphism group, uniformly perfect, commutator subgroup
Received by editor(s): September 12, 2011
Published electronically: November 28, 2012
Additional Notes: The author is partially supported by Grant-in-Aid for Scientific Research (A) 20244003, (S) 24224002, Grant-in-Aid for Exploratory Research 21654009, 24654011, Japan Society for Promotion of Science, and by the Global COE Program at the Graduate School of Mathematical Sciences, University of Tokyo.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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