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

TL;DR

This paper proves that polyhedra with finite cyclic or abelian fundamental groups of rank 1 dominate only finitely many homotopy types, partially answering Borsuk's 1968 question about the finiteness of shapes dominated by such polyhedra.

Contribution

It establishes finiteness results for polyhedra with finite cyclic and rank 1 abelian fundamental groups, extending previous work on finite fundamental groups.

Findings

Polyhedra with finite cyclic fundamental group dominate finitely many homotopy types.

Polyhedra with abelian fundamental group of rank 1 dominate finitely many homotopy types.

Polyhedra dominate finitely many homotopy types of simply connected CW-complexes.

Abstract

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.