In 1968 K. Borsuk asked: Is it true that every finite polyhedron dominates only finitely many different shapes? In this question the notions of shape and shape domination can be replaced by the notions of homotopy type and homotopy domination.We obtained earlier a negative answer to the Borsuk question and next results that the examples of such polyhedra are not rare. In particular, there exist polyhedra with nilpotent fundamental groups dominating infinitely many different homotopy types. On the other hand, we proved that every polyhedron with finite fundamental group dominates only finitely many different homotopy types.Here we obtain next positive results that the same is true for some classes of polyhedra with Abelian fundamental groups and for nilpotent polyhedra. Therefore we also get that every finitely generated, nilpotent torsion-free group has only finitely many r-images up to isomorphism.

Abstract. In 1968 K. Borsuk asked: Does every polyhedron dominate only finitely many different shapes? In this question the notion of shape can be replaced by the notion of homotopy type. We showed earlier that the answer to the Borsuk question is no. However, in a previous paper we proved that every simply connected polyhedron dominates only finitely many different homotopy types (equivalently, shapes). Here we prove that the same is true for polyhedra with finite fundamental group.

Abstract. We show that there exists a finite polyhedron P homotopy dominating infinitely many finite polyhedra K i of different homotopy types such that there isn't any homotopy type between P and K i . This answers negatively the question raised by K. We answer this question showing that there exists even a finite polyhedron with infinitely many left neighbors which are also finite polyhedra.Remark 1. Similarly, in the homotopy category of compact ANR's there were defined the so-called h-neighbors (see [B1]). Since on this category shape theory and homotopy theory coincide, we immediately obtain an example of a finite polyhedron with infinitely many left h-neighbors.Let us note that from the results of Hastings and Heller ([HaHe1],[HaHe2]), there is a 1-1 correspondence between the shapes of compacta and the homotopy types of CW -complexes dominated by a given polyhedron.Hence we will work in the homotopy category of CW -complexes and homotopy classes of maps between them.Remark 2. On the other side, it is easy to find a finite polyhedron with infinitely many right neighbors. It suffices to take a point {p}, then the spheres S i , for