A Weakly Chainable Tree-Like Continuum without the Fixed Point Property

Minc, Piotr

doi:10.1090/s0002-9947-99-02071-1

Cited by 7 publications

(2 citation statements)

References 22 publications

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“…The plane fixed point problem has been listed as the first problem among the classical problems in continuum theory, see Cook, Ingram and Lelek [19], and Prajs and Charatonik [64]. No question has attracted more interest from continua theorists than this problem (see Rogers [65, p. 305]), and many papers connected the topic have appeared since 1930, including [1,2,3,5,6,7,9,12,13,14,15,16,18,21,22,23,24,25,26,27,29,30,31,32,33,36,37,38,39,41,42,46,47,48,50,51,54,55,56,57,58,59,62,63,66,67]. In these papers there were many atte...…”

Section: Introductionmentioning

confidence: 99%

“…The first example of a tree-like continuum without the fixed point property was constructed by Bellamy [7] in 1980 answering a question by Bing [12, Question 2, p 122]. This break-through construction was followed by other examples of tree-like continua without the fixed point property [62,63,55,21,56,57,59,37,36,24]. Applying to those continua and fixed-point-free maps a technique by Fugate and Mohler [23], we get more tree-like continua without the fixed point property.…”

Section: Introductionmentioning

confidence: 99%

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Characterizations of $\mathcal P$-like continua that do not have the fixed point property

Banič¹,

Kennedy²,

Minc³

2021

Preprint

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We give two characterizations of P-like continua X that do not have the fixed point property. Both characterizations are stated in terms of sequences of open covers of X that follow fixed-point-free patterns. We use these to characterize planar tree-like continua that do not have the fixed point property in terms of infinite sequences of tree-chains in the plane that follow fixed-point-free patterns. We also establish a useful relationship between these tree-chains and commutative simplicial diagrams that we use later to construct a finite sequence (of any given length) of tree-chains in the plane that follows a fixed-point-free pattern.An earlier characterization of P-like continua with the fixed point property was given in 1994 by Feuerbacher based on a 1963 result by Mioduszewski. The Mioduszewski-Feuerbacher characterization is expressed in terms of almost commutative inverse diagrams. In contrast, our approach is more geometric, and it may potentially lead to new methods in the elusive search for a planar tree-like continuum without the fixed-point property.

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