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

A Positive Answer to a Question of K. Borsuk on the Capacity of Polyhedra with Finite by Cyclic Fundamental Group