On the Borsuk conjecture concerning homotopy domination

Komendarczyk, Rafal; Kwasik, Sławomir; Rosicki, Witold

doi:10.1007/s40062-015-0125-8

Cited by 1 publication

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“…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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Section: The Depth D(a) Of Anmentioning

confidence: 99%