Contractible -manifolds of finite genus at infinity

Author:
E. M. Brown

Journal:
Trans. Amer. Math. Soc. **245** (1978), 503-514

MSC:
Primary 57N10

MathSciNet review:
511426

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Edward M. Brown,*Proper homotopy theory in simplicial complexes*, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Springer, Berlin, 1974, pp. 41–46. Lecture Notes in Math., Vol. 375. MR**0356041****[2]**E. M. Brown and T. W. Tucker,*On proper homotopy theory for noncompact 3-manifolds*, Trans. Amer. Math. Soc.**188**(1974), 105–126. MR**0334225**, 10.1090/S0002-9947-1974-0334225-X**[3]**J. C. Chipman,*An isomorphism condition for towers of graphs*, Pacific J. Math. (to appear).**[4]**-,*Presentations for proper fundamental groups*, preprint.**[5]**Wolfgang Heil,*On 𝑃²-irreducible 3-manifolds*, Bull. Amer. Math. Soc.**75**(1969), 772–775. MR**0251731**, 10.1090/S0002-9904-1969-12283-4**[6]**J. M. Kister and D. R. McMillan Jr.,*Locally euclidean factors of 𝐸⁴ which cannot be imbedded in 𝐸³*, Ann. of Math. (2)**76**(1962), 541–546. MR**0144322****[7]**D. R. McMillan Jr.,*Some contractible open 3-manifolds*, Trans. Amer. Math. Soc.**102**(1962), 373–382. MR**0137105**, 10.1090/S0002-9947-1962-0137105-X**[8]**D. R. McMillan Jr.,*Cartesian products of contractible open manifolds*, Bull. Amer. Math. Soc.**67**(1961), 510–514. MR**0131280**, 10.1090/S0002-9904-1961-10662-9**[9]**Robert Messer,*Three-dimensional manifolds with finitely generated fundamental groups*, Trans. Amer. Math. Soc.**226**(1977), 119–145. MR**0436149**, 10.1090/S0002-9947-1977-0436149-9**[10]**John Stallings,*On the loop theorem*, Ann. of Math. (2)**72**(1960), 12–19. MR**0121796****[11]**Friedhelm Waldhausen,*On irreducible 3-manifolds which are sufficiently large*, Ann. of Math. (2)**87**(1968), 56–88. MR**0224099****[12]**J. H. C. Whitehead,*A certain open manifold whose group is unity*, Quart. J. Math. Oxford Ser. (2)**6**(1935), 268-279.**[13]**J. H. C. Whitehead,*On 2-spheres in 3-manifolds*, Bull. Amer. Math. Soc.**64**(1958), 161–166. MR**0103473**, 10.1090/S0002-9904-1958-10193-7**[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), 14-22.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57N10

Retrieve articles in all journals with MSC: 57N10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0511426-2

Keywords:
Proper map,
proper homotopy equivalence,
proper fundamental group,
end,
eventually end-irreducible,
open 3-manifold

Article copyright:
© Copyright 1978
American Mathematical Society