There exists a polyhedron with infinitely many left neighbors

Kołodziejczyk, Danuta

doi:10.1090/s0002-9939-00-05812-3

Cited by 7 publications

(10 citation statements)

References 14 publications

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“…The exact computation of capacity of the wedge sum of finitely many spheres with the same or different dimesnions seems interesting. In [15], it has been mentioned that the capacity of k S 1 equals to k + 1, but the proof does not work for n ≥ 2. Kolodziejczyk in [12] asked the following question: Does every polyhedron P with the abelian fundamental group π 1 (P ) dominate only finitely many different homotopy types?…”

Section: The Capacity Of Mooer Spacesmentioning

confidence: 99%

“…". D. Kolodziejczyk in [15] gave a negative answer to this question. However, she investigated some conditions for polyhedra to have finite capacity ( [11,12,13,14]).…”

Section: Introductionmentioning

confidence: 99%

“…Kolodziejczyk in [15] gave a negative answer to this question. She proved that there exist two finite CW-complexes X and Y with dim X = dim Y = 2 such that C(X), C(Y ) and C(X ∩Y ) are finite, while C(X ∪Y ) is infinite.…”

Section: Introductionmentioning

confidence: 99%

“…Kolodziejczyk in [15] showed that there exists a polyhedron (even of dimension 2) homotopy dominates infinitely many polyhedra of different homotopy types, and so she gave a negative answer to this question. Moreover, she proved that such examples are not rare, for every non-abelian poly-Z-group G and an integer n ≥ 3 there exists a polyhedron P with π 1 (P ) ∼ = G and dim P = n dominating infinitely many polyhedra of different homotopy types (see [11]).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the Capacity of Eilenberg-MacLane and Moore Spaces

Mohareri¹,

Mashayekhy²,

Mirebrahimi³

2016

Preprint

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K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum and asked some questions concerning properties of the capacity of compacta. In this paper, we give partial positive answers to three of these questions in some cases. In fact, by describing spaces homotopy dominated by Moore and Eilenberg-MacLane spaces, we obtain the capacity of a Moore space M(A, n) and an Eilenberg-MacLane space K(G, n). Also, we compute the capacity of the wedge sum of finitely many Moore spaces of different degrees and the capacity of the product of finitely many Eilenberg-MacLane spaces of different homotopy types. In particular, we give exact capacity of the wedge sum of finitely many spheres of the same or different dimensions.

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Section: The Capacity Of Mooer Spacesmentioning

confidence: 99%

“…". D. Kolodziejczyk in [15] gave a negative answer to this question. However, she investigated some conditions for polyhedra to have finite capacity ( [11,12,13,14]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Capacity of Eilenberg-MacLane and Moore Spaces

Mohareri¹,

Mashayekhy²,

Mirebrahimi³

2016

Preprint

View full text Add to dashboard Cite

show abstract

“…Borsuk in [5] asked a question: "Is it true that the capacity of every finite polyhedron is finite?". D. Kolodziejczyk in [12] gave a negative answer to this question. Also, she investigated some conditions for polyhedra to have finite capacity ( [13,14]).…”

Section: Introductionmentioning

confidence: 99%

The capacity of some classes of polyhedra

Mohareri¹,

Mashayekhy²,

Mirebrahimi³

2019

HJMS

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K. Borsuk in the seventies introduced the notions of capacity and depth of compacta together with some relevant problems. In this paper, first, we introduce the concepts of the (strong) capacity and the (strong) depth of an object in an arbitrary category. Then in the category of groups, we compute the (strong) capacity and the (strong) depth of some well-known groups. Finally, we find an upper bound for the depth of some classes of finite polyhedra which generalizes a result of D. Kolodziejczyk in this subject.The previous definition of the capacity of a compatum in the shape category of compacta coincides with Borsuk's definiton of the capacity (see [3]).2. The depth D(A) of an A ∈ Obj C is the least upper bound of the lengths of all chains for A. If this upper bound is infinite, we write D(A) = N 0 .3. A chain X k < s · · · < s X 1 d A, where X i ∈ Obj C for i = 1, · · · , k, is called an s-chain of length k for A ∈ Obj C. 4. The strong depth SD(A) of A is the least upper bound of the lengths of all s-chains for A. If this upper bound is infinite, we write SD(A) = N 0 .Note that our definition of strong depth of a compactum in the shape category of compacta coincides with Borsuk's definition of the depth of a compactum (for more details, see [3]).

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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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There exists a polyhedron with infinitely many left neighbors

Cited by 7 publications

References 14 publications

On the Capacity of Eilenberg-MacLane and Moore Spaces

On the Capacity of Eilenberg-MacLane and Moore Spaces

The capacity of some classes of polyhedra

On the capacity and depth of compact surfaces

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