On proper homotopy theory for noncompact 3-manifolds

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 × [ 0 , 1 ] F \times [0,1] or a half-open product F × [ 0 , 1 ) F \times [0,1) where F is a 2-manifold is actually homeomorphic to F × [ 0 , 1 ] F \times [0,1] or F × [ 0 , 1 ) 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.