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

Kołodziejczyk, Danuta

doi:10.1016/j.topol.2003.09.016

Cited by 7 publications

(4 citation statements)

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“…The following corollary is well known to the experts [17,11,16] in this research area: Corollary 1.3. Suppose X and Y are compact, h-equal ANR spaces with the polycyclic-byfinite fundamental groups, then X and Y is a Hopfian pair.…”

Section: Introductionmentioning

confidence: 93%

On the Borsuk conjecture concerning homotopy domination

Komendarczyk¹,

Kwasik²,

Rosicki³

2015

J. Homotopy Relat. Struct.

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Abstract. In the seminal monograph Theory of retracts, Borsuk raised the following question: suppose two compact ANR spaces are h-equal, i.e. mutually homotopy dominate each other, are they homotopy equivalent? The current paper approaches this question in two ways. On one end, we provide conditions on the fundamental group which guarantee a positive answer to the Borsuk question. On the other end, we construct various examples of compact h-equal, not homotopy equivalent continua, with distinct properties. The first class of these examples has trivial all known algebraic invariants (such as homology, homotopy groups etc.) The second class is given by n-connected continua, for any n, which are infinite CW -complexes, and hence ANR spaces, on a complement of a point.

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Section: Introductionmentioning

confidence: 93%

On the Borsuk conjecture concerning homotopy domination

Komendarczyk¹,

Kwasik²,

Rosicki³

2015

J. Homotopy Relat. Struct.

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“…Note that by [13,Theorem 3], if A ∈ AN R and π 1 (A) is virtually polycyclic, then every X d A has d-equality property.…”

Section: Strong Capacity and Strong Depth In A Categorymentioning

confidence: 99%

“…By [13,Theorem 3], if X ∈ AN R and π 1 (X) is virtually polycyclic fundamental group, then X has d-equality property (also see [19]). Also, every Hopfian group has d-equality property (see Prposition 3.2).…”

Section: The Depth D(a) Of Anmentioning

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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“…For polyhedra with polycyclic-by-finite fundamental groups the answers are also positive (see [K,Theorem 3,Theorem 5]). Thus, one may ask how about 2dimensional polyhedra, in particular 2-dimensional polyhedra with soluble fundamental groups.…”

Section: Introductionmentioning

confidence: 99%