Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Small sets of homeomorphisms which control manifolds

Author: Norman Levitt
Journal: Proc. Amer. Math. Soc. 57 (1976), 173-178
MSC: Primary 57D25; Secondary 49E15
MathSciNet review: 0405450
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {N^n}$ be a smooth connected paracompact manifold without boundary. A set $ D$ of self-homeomorphisms of $ {M^n}$ to itself is called controllable iff the semigroup generated by $ D$ acts transitively on $ {M^n}$.

Theorem A. There is a complete vector field $ X$ on $ {M^n}$ and a self-homeomorphism $ H$ so that the set $ D$ consisting of $ H,{H^{ - 1}}$ and $ {X_t},t \in {\mathbf{R}}$, is controllable.

Theorem B. Let $ n \ne 4$ and let $ {M^n}$ be compact and orientable. If $ n$ is even, $ \geqslant 6$, let $ {M^n}$ be simply connected. If $ n \equiv 0(4)$, let signature $ {M^n} = 0$. Then there is a vector field $ X$ on $ {M^n}$ and a self-homeomorphism $ H$ so that the set consisting of $ H,{H^{ - 1}}$ and $ {X_t},t \geqslant 0$, is controllable.

References [Enhancements On Off] (What's this?)

  • [1] D. Barden, The structure of manifolds, Thesis, Cambridge, 1963.
  • [2] Jean Cerf, Topologie de certains espaces de plongements, Bull. Soc. Math. France 89 (1961), 227–380 (French). MR 0140120
  • [3] F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. Die Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. MR 0202713
  • [4] N. Levitt, Applications of engulfing, Thesis, Princeton University, 1967.
  • [5] Norman Levitt and Héctor J. Sussmann, On controllability by means of two vector fields, SIAM J. Control 13 (1975), no. 6, 1271–1281. MR 0402812
  • [6] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331
  • [7] -, Lectures on the $ h$-cobordism theorem, Princeton Univ. Press, Princeton, N.J., 1965. MR 32 #8352.
  • [8] Richard S. Palais, Extending diffeomorphisms, Proc. Amer. Math. Soc. 11 (1960), 274–277. MR 0117741, 10.1090/S0002-9939-1960-0117741-0
  • [9] H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, Leipzig, 1934; reprint, Chelsea, New York, 1947.
  • [10] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR 0153022
  • [11] H. E. Winkelnkemper, Thesis, Princeton University, 1970.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57D25, 49E15

Retrieve articles in all journals with MSC: 57D25, 49E15

Additional Information

Keywords: Controllable set of vector fields, controllable set of homeomorphisms, gradient-like vector field, topologically conjugate vector fields, twisted double theorem
Article copyright: © Copyright 1976 American Mathematical Society