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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On proper homotopy theory for noncompact $ 3$-manifolds

Authors: E. M. Brown and T. W. Tucker
Journal: Trans. Amer. Math. Soc. 188 (1974), 105-126
MSC: Primary 57A65
MathSciNet review: 0334225
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Abstract: Proper homotopy groups analogous to the usual homotopy groups are defined. They are used to prove, modulo the Poincaré conjecture, that a noncompact 3-manifold having the proper homotopy type of a closed product $ F \times [0,1]$ or a half-open product $ F \times [0,1)$ where F is a 2-manifold is actually homeomorphic to $ F \times [0,1]$ or $ F \times [0,1)$, respectively. By defining a concept for noncompact manifolds similar to boundary-irreducibility, a well-known result of Waldhausen concerning homotopy and homeomorphism type of compact 3-manifolds is extended to the noncompact case.

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Keywords: Proper map, end, proper homotopy groups, incompressible surface, $ {p^2}$-irreducible, boundary-irreducible, end-irreducible
Article copyright: © Copyright 1974 American Mathematical Society