Proper homotopy groups analogous to the usual homotopy groups are defined. They are used to prove, modulo the Poincare conjecture, that a noncompact 3-manifold having the proper homotopy type of a closed product F x [0, 1] or a half-open product F x [0, 1) where F is a 2-manifold is actually homeomorphic to F x [0, 1] or F x [0, 1), respectively. By defining a concept for noncompact manifolds similar to boundary-irreducibility, a wellknown result of Waldhausen concerning homotopy and homeomorphism type of compact 3-manifolds is extended to the noncompact case. Introduction. There have been several papers in recent years on noncompact manifolds, proving theorems having homotopy-theoretical hypotheses (particularly "at infinity") and having homeomorphism type conclusions. Some examples are Husch and Price [81, Edwards [61, Siebenmann [121 and [131, Levine et al. [2]. This paper is another of the same. Siebenmann's paper [121 argues strongly the case that for noncompact spaces the homotopy hypotheses should be on the category of proper maps rather than the category of all continuous maps. We recall that a continuous map f: X Y is proper if f 1(C) is compact whenever C is compact. In this paper we shall work in the category of locally finite simplicial complexes and proper maps, and we shall be concerned with proving theorems about noncompact 3-manifolds. Our hypotheses will be in terms of functors which are invariants of proper homotopy type (plus a possibly redundant one to get around the 3-dimensional Poincare' conjecture). We remark that a proper homotopy is a homotopy which is proper as a single function of two variables (which is not the same as a continuous family ft of proper maps). In addition to the usual homotopy groups, we shall use two functors which, while invariants of proper homotopy type, are not invariants of the usual (continuous) homotopy type. The first is well known, it is the space Received by the editors March 17, 1971 and, in revised form, September 20, 1972 and January 22, 1973. AMS (MOS) subject classifications (1970). Primary 57C99, 57A99, 55A99.
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