Polyhedra dominating finitely many different homotopy types

Abstract

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.

On the capacity and depth of compact surfaces

“…". D. Kolodziejczyk in [13] gave a negative answer to this question. Also, in [8] she proved that there exist polyhedra with infinite capacity and finite depth.…”

“…Also, in [8] she proved that there exist polyhedra with infinite capacity and finite depth. Moreover, she investigated some conditions for polyhedra to have finite capacity ( [9,10,12]). For instance, polyhedra with finite fundamental groups and polyhdera P with abelian fundamental groups π 1 (P ) and finitely generated homology groups H i (P ), for i ≥ 2, have finite capacity.…”

The capacity of some classes of polyhedra

“…The answers to most of problems of [3] are given by D. Kolodziejczyk in [12]. In [18], answering the question (4) of [3]: "Is it true that the capacity of every finite polyhedron is finite? ", Kolodziejczyk showed that there is a polyhedron dominating infinitely many different homotopy types of polyhedra.…”

The capacity of wedge sum of spheres of different dimensions

“…". D. Kolodziejczyk in [9] gave a negative answer to this question. Also, she investigated some conditions for polyhedra to have finite capacity ( [5,6,7,8]).…”

“…D. Kolodziejczyk in [9] gave a negative answer to this question. Also, she investigated some conditions for polyhedra to have finite capacity ( [5,6,7,8]). For instance, a polyhedron Q with finite fundamental group π 1 (Q) and a polyhedron P with abelian fundamental group π 1 (P ) and finitely generated homology groups H i (P ), for i ≥ 2 whereP is the universal cover of P , have finite capacities.…”