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

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

Published electronically:
October 6, 1999

MathSciNet review:
1641077

Full-text PDF

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,*Open 3-manifolds which are simply connected at infinity*, Proc. Amer. Math. Soc.**14**(1963), 391-395. MR**27:732****3.**R. D. Edwards and R. C. Kirby,*Deformations of spaces of imbeddings*, Ann. Math. (2)**93**(1971), 63-88. MR**44:1032****4.**M. H. Freedman,*The topology of four-dimensional manifolds*, J. Differential Geom.**17**(1982), 357-453. MR**84b:57006****5.**M. H. Freedman and F. Quinn,*Topology of 4-manifolds.*Princeton Mathematical Series 39, Princeton University Press, Princeton, NJ, 1990. MR**94b:57021****6.**M. Freedman and R. Skora,*Strange actions of groups on spheres*, J. Differential Geometry**25**(1987), 75-98. MR**88a:57074****7.**R. Geoghegan and M. L. Mihalik,*The fundamental group at infinity*, Topology**35**(1996), 655-669. MR**97h:57002****8.**J. Hempel,*3-Manifolds*, Ann. of Math. Studies, No. 86, Princeton, 1976. MR**54:3702****9.**R. C. Kirby and L. C. Siebenmann,*Foundational essays on topological manifolds, smoothings, and triangulations. With notes by John Milnor and Michael Atiyah.*Annals of Mathematics Studies, No. 88. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. MR**58:31082****10.**W. Massey,*Algebraic Topology: An Introduction*, Graduate Texts in Mathematics No. 56, Springer-Verlag, 1977. MR**35:2271****11.**J. Milnor,*On spaces having the homotopy type of a CW-complex*, Trans. Amer. Math. Soc.**90**(1959), 272-280. MR**20:6700****12.**E. E. Moise,*Affine structures in -manifolds. V. The triangulation theorem and Hauptvermutung.*Ann. of Math. (2)**56**, (1952), 96-114. MR**14:72d****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**38:6601****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**31:6218****17.**D. G. Wright,*Contractible open manifolds which are not covering spaces*, Topology**31**(1992), 281-291. MR**93f:57004**

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:
https://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