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

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Abstract: Let be a smooth connected paracompact manifold without boundary. A set of self-homeomorphisms of to itself is called controllable iff the semigroup generated by acts transitively on .

Theorem A. *There is a complete vector field on and a self-homeomorphism so that the set consisting of and , is controllable*.

Theorem B. *Let and let be compact and orientable. If is even, , let be simply connected. If , let signature . Then there is a vector field on and a self-homeomorphism so that the set consisting of and , is controllable*.

**[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 -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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0405450-1

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