Contractible 3-manifolds of finite genus at infinity

Brown, E. M.

doi:10.1090/S0002-9947-1978-0511426-2

Contractible $3$-manifolds of finite genus at infinity
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Trans. Amer. Math. Soc. 245 (1978), 503-514 Request permission

Abstract:

A class of contractible open 3-manifolds is defined. It is shown that all contractible open 3-manifolds 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.

References

Edward M. Brown, Proper homotopy theory in simplicial complexes, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974, pp. 41–46. MR 0356041
E. M. Brown and T. W. Tucker, On proper homotopy theory for noncompact $3$-manifolds, Trans. Amer. Math. Soc. 188 (1974), 105–126. MR 334225, DOI 10.1090/S0002-9947-1974-0334225-X

An isomorphism condition for towers of graphs

Presentations for proper fundamental groups

Wolfgang Heil, On $P^{2}$-irreducible $3$-manifolds, Bull. Amer. Math. Soc. 75 (1969), 772–775. MR 251731, DOI 10.1090/S0002-9904-1969-12283-4
J. M. Kister and D. R. McMillan Jr., Locally euclidean factors of $E^{4}$ which cannot be imbedded in $E^{3}$, Ann. of Math. (2) 76 (1962), 541–546. MR 144322, DOI 10.2307/1970374
D. R. McMillan Jr., Some contractible open $3$-manifolds, Trans. Amer. Math. Soc. 102 (1962), 373–382. MR 137105, DOI 10.1090/S0002-9947-1962-0137105-X
D. R. McMillan Jr., Cartesian products of contractible open manifolds, Bull. Amer. Math. Soc. 67 (1961), 510–514. MR 131280, DOI 10.1090/S0002-9904-1961-10662-9
Robert Messer, Three-dimensional manifolds with finitely generated fundamental groups, Trans. Amer. Math. Soc. 226 (1977), 119–145. MR 436149, DOI 10.1090/S0002-9947-1977-0436149-9
John Stallings, On the loop theorem, Ann. of Math. (2) 72 (1960), 12–19. MR 121796, DOI 10.2307/1970146
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594

A certain open manifold whose group is unity

6

J. H. C. Whitehead, On $2$-spheres in $3$-manifolds, Bull. Amer. Math. Soc. 64 (1958), 161–166. MR 103473, DOI 10.1090/S0002-9904-1958-10193-7

On the group of a certain linkage

8