Dilations on invertible spaces
Author:
Ellard Nunnally
Journal:
Trans. Amer. Math. Soc. 123 (1966), 437448
MSC:
Primary 54.80
MathSciNet review:
0208575
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Abstract: This paper primarily concerns certain groups of homeomorphisms which are associated in a natural way with a variety of spaces, which satisfy a set of axiomatic conditions put forth in §1. Let us suppose that is a space of the type in question and that is an appropriate group of homeomorphisms of onto itself. In §2 we demonstrate the existence of a nonvoid subcollection , the ``topological dilations,'' of which is characterized in Theorem 1 in the following fashion: suppose and , then if and only if is a conjugate of , that is if and only if there exists an element of such that . We proceed then to show in §3 that if and are nonidentity elements of , then we may find such that the product . We then combine this fact with the characterization of mentioned above to conclude that each element of is a ``universal'' element of in the sense that if , then any element of may be represented as the product of two conjugates of . Furthermore we conclude that if is not the identity element of , then can be represented as the product of three conjugates of any nonidentity element of . Finally, we apply the conclusions to groups of homeomorphisms of certain spaces: for example spheres, cells, the Cantor set, etc.
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 R. D. Anderson, The three conjugates theorem, Topology (to appear).
 [2]
 , The algebraic simplicity of certain groups of homeomorphisms, Amer. J. Math., 80 (1958), 955963. MR 0098145 (20:4607)
 [3]
 Morton Brown and Herman Gluck, Stable structures on manifolds. I, Ann. of Math. 79 (1964), 117. MR 0158383 (28:1608a)
 [4]
 Gordon Fisher, On the group of all homeomorphisms of a manifold, Trans. Amer. Math. Soc. 97 (1960), 193212. MR 0117712 (22:8487)
 [5]
 J. M. Kister, Isotopies in manifolds, Trans. Amer. Math. Soc. 97 (1960), 213224. MR 0120628 (22:11378)
 [6]
 H. Kneser, Die Deformationssatz der Einfach Zusammenhangenden Flachen, Math. Z. 25 (1926), 362372. MR 1544816
 [7]
 S. Ulam and J. von Neumann, On the group of homeomorphisms of the surface of a sphere, Abstract 283, Bull. Amer. Math. Soc. 53 (1947), 506.
 [8]
 N. J. Fine and G. E. Schweigert, On the group of homeomorphisms of an arc, Ann. of Math. 62 (1955), 237253. MR 0072460 (17:288b)
 [9]
 M. E. Hamstrom and E. Dyer, Regular mappings and the space of homeomorphisms on a manifold, Duke Math. J. 25 (1958), 521531. MR 0096202 (20:2695)
 [10]
 M. E. Hamstrom, Regular mappings and the space of homeomorphisms on a manifold, Abstract 56439, Notices Amer. Math. Soc. 6 (1959), 783784.
 [11]
 J. H. Roberts, Local arcwise connectivity in the space of homeomorphisms of onto itself, Summary of Lectures, Summer Institute on Set Theoretic Topology, Madison, Wisconsin, 1955; p. 100.
 [12]
 M. K. Fort, Jr., A proof that the group of homeomorphisms of the plane onto itself is locally arcwise connected, Proc. Amer. Math. Soc. 1 (1950), 5962. MR 0033017 (11:381g)
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DOI:
http://dx.doi.org/10.1090/S00029947196602085757
PII:
S 00029947(1966)02085757
Article copyright:
© Copyright 1966
American Mathematical Society
