Dilations on invertible spaces

Author:
Ellard Nunnally

Journal:
Trans. Amer. Math. Soc. **123** (1966), 437-448

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.

**[1]**R. D. Anderson,*The three conjugates theorem*, Topology (to appear).**[2]**R. D. Anderson,*The algebraic simplicity of certain groups of homeomorphisms*, Amer. J. Math.**80**(1958), 955–963. MR**0098145****[3]**Morton Brown and Herman Gluck,*Stable structures on manifolds. I. Homeomorphisms of 𝑆ⁿ*, Ann. of Math. (2)**79**(1964), 1–17. MR**0158383****[4]**Gordon M. Fisher,*On the group of all homeomorphisms of a manifold*, Trans. Amer. Math. Soc.**97**(1960), 193–212. MR**0117712**, 10.1090/S0002-9947-1960-0117712-9**[5]**J. M. Kister,*Isotopies in 3-manifolds*, Trans. Amer. Math. Soc.**97**(1960), 213–224. MR**0120628**, 10.1090/S0002-9947-1960-0120628-5**[6]**Hellmuth Kneser,*Die Deformationssätze der einfach zusammenhängenden Flächen*, Math. Z.**25**(1926), no. 1, 362–372 (German). MR**1544816**, 10.1007/BF01283844**[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. (2)**62**(1955), 237–253. MR**0072460****[9]**Mary-Elizabeth Hamstrom and Eldon Dyer,*Regular mappings and the space of homeomorphisms on a 2-manifold*, Duke Math. J.**25**(1958), 521–531. MR**0096202****[10]**M. E. Hamstrom,*Regular mappings and the space of homeomorphisms on a -manifold*, Abstract 564-39, Notices Amer. Math. Soc.**6**(1959), 783-784.**[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 all homeomorphisms of the plane onto itself is locally arcwise connected*, Proc. Amer. Math. Soc.**1**(1950), 59–62. MR**0033017**, 10.1090/S0002-9939-1950-0033017-8

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1966-0208575-7

Article copyright:
© Copyright 1966
American Mathematical Society