Polyhedra with virtually polycyclic fundamental groups have finite depth

Abstract

Abstract. The notions of capacity and depth of compacta were introduced by K. Borsuk in the seventies together with some open questions. In a previous paper, in connection with one of them, we proved that there exist polyhedra with polycyclic fundamental groups and infinite capacity, i.e. dominating infinitely many different homotopy types (or equivalently, shapes). In this paper we show that every polyhedron with virtually polycyclic fundamental group has finite depth, i.e., there is a bound on the lengths of all descending sequences of different homotopy types (or shapes) dominated by this polyhedron. As a corollary, we deduce that for two ANR's with virtually polycyclic fundamental groups the so-called index of h-proximity, introduced by K. Borsuk in his monograph on retract theory, is finite. We also obtain an answer to some question of K. Borsuk concerning homotopy (or shape) decompositions of polyhedra into simple constituents.

The capacity of some classes of polyhedra

“…Recall that a domination in a given category C is a morphism g : Y −→ X, X, Y ∈ ObjC, for which there exists a morphism f : X −→ Y of C such that g • f = id X . In this case, we say that X is dominated by Y and we write X d Y (rather than the notation " ", see for example [17]). Also, X is called strongly dominated by Y and is denoted by X < s Y if X d Y holds but Y d X fails.…”

“…In this section, we generalize a theorem of Kolodziejczyk (see [17,Theorem 2]) which states that every polyhedron with virtually polycyclic fundamental group has finite depth. Note that for polyhedra, the notions shape and shape domination can be replaced by the notions homotopy type and homotopy domination, respectively (see [15]).…”

“…Accordingly, the examples of polyhedra with infinite capacity are even frequent. In [17], Kolodziejczyk also proved that every polyhedron with virtually polycyclic fundamental group has finite depth.…”

“…Also, we show that every virtually polycyclic group has finite depth. Finally, in Section 4, we generalize a theorem of Kolodziejczyk (see [17,Theorem 2]) which states that every polyhedron with virtually polycyclic fundamental group has finite depth. Also, we present an upper bound for the depth of some classes of polyhedra.…”

On the capacity and depth of compact surfaces

“…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. Also, in [11] she proved that polyhedra with virtually polycyclic fundamental group have finite depth.…”