Polyhedra with finite fundamental group dominate finitely many different homotopy types

Kołodziejczyk, Danuta

doi:10.4064/fm180-1-1

Cited by 7 publications

(18 citation statements)

References 4 publications

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“…We now extend the methods used in the case of polyhedra with finite fundamental groups (see [14]) to some classes of polyhedra with Abelian fundamental groups.…”

Section: Polyhedra With Abelian Fundamental Groupsmentioning

confidence: 98%

“…In the proof of Theorem 2 we will use the following three lemmas (proofs of Lemmas 2 and 3, and more about Lemma 1 with Corollary 2 can be found in [14]). …”

Section: Definitionmentioning

confidence: 99%

“…Theorem 1 of [14] states that for every polyhedron P , all the CW-complexes X P can be partitioned into finitely many classes such that for any two CW-complexes belonging to the same class, there is a homology equivalence between them.…”

Section: Theorem 1 Every Nilpotent Polyhedron Dominates Only Finitelmentioning

confidence: 99%

“…In [14] we proved, in an other way, that polyhedra with finite fundamental groups dominate only finitely many different homotopy types.…”

mentioning

confidence: 94%

“…In this paper, extending the methods of [14], we prove that for some classes of polyhedra with Abelian fundamental group, the answer to the Borsuk question is positive.…”

mentioning

confidence: 95%

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Polyhedra dominating finitely many different homotopy types

Kołodziejczyk

2005

Topology and its Applications

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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.

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“…We now extend the methods used in the case of polyhedra with finite fundamental groups (see [14]) to some classes of polyhedra with Abelian fundamental groups.…”

Section: Polyhedra With Abelian Fundamental Groupsmentioning

confidence: 98%

“…In the proof of Theorem 2 we will use the following three lemmas (proofs of Lemmas 2 and 3, and more about Lemma 1 with Corollary 2 can be found in [14]). …”

Section: Definitionmentioning

confidence: 99%

Section: Theorem 1 Every Nilpotent Polyhedron Dominates Only Finitelmentioning

confidence: 99%

“…In [14] we proved, in an other way, that polyhedra with finite fundamental groups dominate only finitely many different homotopy types.…”

mentioning

confidence: 94%

“…In this paper, extending the methods of [14], we prove that for some classes of polyhedra with Abelian fundamental group, the answer to the Borsuk question is positive.…”

mentioning

confidence: 95%

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Polyhedra dominating finitely many different homotopy types

Kołodziejczyk

2005

Topology and its Applications

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On the capacity and depth of compact surfaces

Abbasi

Mashayekhy

2020

J. Homotopy Relat. Struct.

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K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of capacity and depth of a compactum. In this paper, we compute the capacity and depth of compact surfaces. We show that the capacity and depth of every compact orientable surface of genus g ≥ 0 is equal to g + 2. Also, we prove that the capacity and depth of a compact non-orientable surface of genus g > 0 is [ g 2 ] + 2.

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Homotopy dominations by polyhedra with polycyclic-by-finite fundamental groups

Kołodziejczyk

2005

Topology and its Applications

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In the previous papers, in connection with a question of K. Borsuk, we proved that there exist polyhedra with polycyclic fundamental groups homotopy dominating infinitely many different homotopy types. Here we consider a few problems of K. Borsuk concerning infinite chains of polyhedra or FANR's ordered by the relation of domination (in homotopy or shape category) and obtain that for polyhedra with polycyclic-by-finite fundamental groups, there are no pathology similar to the above.

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Polyhedra with finite fundamental group dominate finitely many different homotopy types

Cited by 7 publications

References 4 publications

Polyhedra dominating finitely many different homotopy types

Polyhedra dominating finitely many different homotopy types

On the capacity and depth of compact surfaces

Homotopy dominations by polyhedra with polycyclic-by-finite fundamental groups

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