  Karol Borsuk in 1968 asked: Is it true that every finite polyhedron dominates
only finitely many different shapes? Danuta Kolodziejczyk showed that generally
an answer to the Borsuk question is negative and also presented a positive
answer by proving that every polyhedron with finite fundamental group dominates
only finitely many different homotopy types (hence shapes). In this paper, we
show that polyhedra with finite by cyclic fundamental group dominate only
finitely many homotopy types. As a consequence, we give a partial positive
answer to this question of Kolodziejczyk: Does every polyhedron with abelian
fundamental group dominate only finitely many different homotopy types? In
fact, we that every polyhedron with abelian fundamental group of rank 1
dominates only finitely many different homotopy types. Finally, we prove that
every polyhedron dominates only finitely many homotopy types of simply
connected CW-complexes.
