On covering translations and homeotopy groups of contractible open -manifolds

Author:
Robert Myers

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1563-1566

MSC (1991):
Primary 57M10; Secondary 57N10, 57N13, 57N15, 57N37, 57M60, 57S30

Published electronically:
October 6, 1999

MathSciNet review:
1641077

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives a new proof of a result of Geoghegan and Mihalik which states that whenever a contractible open -manifold which is not homeomorphic to is a covering space of an -manifold and either or and is irreducible, then the group of covering translations injects into the homeotopy group of .

**1.**A. V. Chernavskii,*Local contractibility of the group of homeomorphisms of a manifold*, Math. USSR-Sb.,**8**(1969), 287-333.**2.**C. H. Edwards Jr.,*Open 3-manifolds which are simply connected at infinity*, Proc. Amer. Math. Soc.**14**(1963), 391–395. MR**0150745**, 10.1090/S0002-9939-1963-0150745-3**3.**Robert D. Edwards and Robion C. Kirby,*Deformations of spaces of imbeddings*, Ann. Math. (2)**93**(1971), 63–88. MR**0283802****4.**Michael Hartley Freedman,*The topology of four-dimensional manifolds*, J. Differential Geom.**17**(1982), no. 3, 357–453. MR**679066****5.**Michael H. Freedman and Frank Quinn,*Topology of 4-manifolds*, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR**1201584****6.**Michael H. Freedman and Richard Skora,*Strange actions of groups on spheres*, J. Differential Geom.**25**(1987), no. 1, 75–98. MR**873456****7.**Ross Geoghegan and Michael L. Mihalik,*The fundamental group at infinity*, Topology**35**(1996), no. 3, 655–669. MR**1396771**, 10.1016/0040-9383(95)00033-X**8.**John Hempel,*3-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619****9.**Robion C. Kirby and Laurence C. Siebenmann,*Foundational essays on topological manifolds, smoothings, and triangulations*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah; Annals of Mathematics Studies, No. 88. MR**0645390****10.**William S. Massey,*Algebraic topology: An introduction*, Harcourt, Brace & World, Inc., New York, 1967. MR**0211390****11.**John Milnor,*On spaces having the homotopy type of a 𝐶𝑊-complex*, Trans. Amer. Math. Soc.**90**(1959), 272–280. MR**0100267**, 10.1090/S0002-9947-1959-0100267-4**12.**Edwin E. Moise,*Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung*, Ann. of Math. (2)**56**(1952), 96–114. MR**0048805****13.**R. Myers,*Contractible open 3-manifolds which non-trivially cover only non-compact 3-manifolds*, Topology**38**, (1999), 85-94. CMP**98:17****14.**R. Myers,*Contractible open 3-manifolds with free covering translation groups*, Topology Appl., to appear.**15.**L. C. Siebenmann,*On detecting Euclidean space homotopically among topological manifolds.*, Invent. Math.**6**(1968), 245–261. MR**0238325****16.**C. T. C. Wall,*Open 3-manifolds which are 1-connected at infinity*, Quart. J. Math. Oxford Ser. (2)**16**(1965), 263–268. MR**0181993****17.**David G. Wright,*Contractible open manifolds which are not covering spaces*, Topology**31**(1992), no. 2, 281–291. MR**1167170**, 10.1016/0040-9383(92)90021-9

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
57M10,
57N10,
57N13,
57N15,
57N37,
57M60,
57S30

Retrieve articles in all journals with MSC (1991): 57M10, 57N10, 57N13, 57N15, 57N37, 57M60, 57S30

Additional Information

**Robert Myers**

Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Email:
myersr@math.okstate.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05163-1

Keywords:
Contractible open manifold,
covering space,
homeotopy group,
mapping class group

Received by editor(s):
October 17, 1997

Received by editor(s) in revised form:
July 10, 1998

Published electronically:
October 6, 1999

Additional Notes:
Research at MSRI is supported in part by NSF grant DMS-9022140.

Communicated by:
Ronald A. Fintushel

Article copyright:
© Copyright 2000
American Mathematical Society