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Polyhedra with virtually polycyclic fundamental groups have finite depth

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Kołodziejczyk D.

Fundamenta Mathematicae

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2007

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tom 197

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nr 1

229-238

EN

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.

2

There exists a polyhedron dominating infinitely many different homotopy types

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Kołodziejczyk D.

Fundamenta Mathematicae

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1996

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tom 151

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nr 1

39-46

3

Polyhedra with finite fundamental group dominate finitely many different homotopy types

100%

Kołodziejczyk D.

Fundamenta Mathematicae

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2003

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tom 180

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nr 1

1-9

EN

In 1968 K. Borsuk asked: Does every polyhedron dominate only finitely many different shapes? In this question the notion of shape can be replaced by the notion of homotopy type. We showed earlier that the answer to the Borsuk question is no. However, in a previous paper we proved that every simply connected polyhedron dominates only finitely many different homotopy types (equivalently, shapes). Here we prove that the same is true for polyhedra with finite fundamental group.

4

Homotopy dominations within polyhedra

100%

Kołodziejczyk D.

Fundamenta Mathematicae

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2003

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tom 178

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nr 3

189-202

EN

We show the existence of a finite polyhedron P dominating infinitely many different homotopy types of finite polyhedra and such that there is a bound on the lengths of all strictly descending sequences of homotopy types dominated by P. This answers a question of K. Borsuk (1979) dealing with shape-theoretic notions of "capacity" and "depth" of compact metric spaces. Moreover, π₁(P) may be any given non-abelian poly-ℤ-group and dim P may be any given integer n ≥ 3.

Ograniczanie wyników

4 Fundamenta Mathematicae

4 Kołodziejczyk D.

1 2007

2 2003

1 1996

Polyhedra with virtually polycyclic fundamental groups have finite depth

There exists a polyhedron dominating infinitely many different homotopy types

Polyhedra with finite fundamental group dominate finitely many different homotopy types

Homotopy dominations within polyhedra