On the Proper Homotopy Invariance of the Tucker Property

A non-compact polyhedron P is Tucker if, for any compact subset K . P, the fundamental group . 1(P . K) is finitely generated. The main result of this note is that a manifold which is proper homotopy equivalent to a Tucker polyhedron is Tucker. We use Poenaru.s theory of the equivalence relations forced by the singularities of a non-degenerate simplicial map.