Contractible manifolds of finite genus at infinity
Author:
E. M. Brown
Journal:
Trans. Amer. Math. Soc. 245 (1978), 503514
MSC:
Primary 57N10
MathSciNet review:
511426
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Abstract: A class of contractible open 3manifolds is defined. It is shown that all contractible open 3manifolds which can be written as a union of cubes with a bounded number of handles are in this class. It is shown that a proper map between manifolds of this class which induces an isomorphism of proper fundamental groups (e.g. a proper homotopy equivalence) is proper homotopic to a homeomorphism. A naturality condition for homomorphisms of proper fundamental groups is developed. It is shown that a natural homomorphism between the proper fundamental groups of these manifolds is induced by a proper map.
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 E. M. Brown and T. W. Tucker, On proper homotopy theory for noncompact 3manifolds, Trans. Amer. Math. Soc. 188 (1974), 105126. MR 0334225 (48:12544)
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 J. C. Chipman, An isomorphism condition for towers of graphs, Pacific J. Math. (to appear).
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 , Presentations for proper fundamental groups, preprint.
 [5]
 W. Heil, On irreducible 3manifolds, Bull. Amer. Math. Soc. 75 (1969), 772775. MR 0251731 (40:4958)
 [6]
 J. M. Kister and D. R. McMillan, Jr., Locally Euclidean factors of which cannot be imbedded in , Ann. of Math. (2) 76 (1962), 541546. MR 0144322 (26:1868)
 [7]
 D. R. McMillan, Jr., Some contractible open 3manifolds, Trans. Amer. Math. Soc. 102 (1962), 373382. MR 0137105 (25:561)
 [8]
 , Cartesian products of contractible open 3manifolds, Bull. Amer. Math. Soc. 67 (1961), 510514. MR 0131280 (24:A1132)
 [9]
 R. A. Messer, Three dimensional manifolds with finitely generated fundamental groups, Trans. Amer. Math. Soc. 226 (1977), 119145. MR 0436149 (55:9099)
 [10]
 J. Stallings, On the loop theorem, Ann. of Math. (2) 72 (1960), 1219. MR 0121796 (22:12526)
 [11]
 F. Waldhausen, On irreducible 3manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 5688. MR 0224099 (36:7146)
 [12]
 J. H. C. Whitehead, A certain open manifold whose group is unity, Quart. J. Math. Oxford Ser. (2) 6 (1935), 268279.
 [13]
 , On 2spheres in 3manifolds, Bull. Amer. Math. Soc. 64 (1958), 161166. MR 0103473 (21:2241)
 [14]
 J. H. C. Whitehead and M. H. A. Newman, On the group of a certain linkage, Quart. J. Math. Oxford Ser. (2) 8 (1937), 1422.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197805114262
PII:
S 00029947(1978)05114262
Keywords:
Proper map,
proper homotopy equivalence,
proper fundamental group,
end,
eventually endirreducible,
open 3manifold
Article copyright:
© Copyright 1978
American Mathematical Society
