On proper homotopy theory for noncompact 3-manifolds

Brown, E. M.; Tucker, T. W.

doi:10.1090/S0002-9947-1974-0334225-X

On proper homotopy theory for noncompact $3$-manifolds
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by E. M. Brown and T. W. Tucker

Trans. Amer. Math. Soc. 188 (1974), 105-126

DOI: https://doi.org/10.1090/S0002-9947-1974-0334225-X

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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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