The fixed-point property for deformations of tree-like continua

Charles Hagopian

Fundamenta Mathematicae (1998)

  • Volume: 155, Issue: 2, page 161-176
  • ISSN: 0016-2736

Abstract

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Let f be a map of a tree-like continuum M that sends each arc-component of M into itself. We prove that f has a fixed point. Hence every tree-like continuum has the fixed-point property for deformations (maps that are homotopic to the identity). This result answers a question of Bellamy. Our proof resembles an old argument of Brouwer involving uncountably many tangent curves. The curves used by Brouwer were originally defined by Peano. In place of these curves, we use rays that were originally defined by Borsuk.

How to cite

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Hagopian, Charles. "The fixed-point property for deformations of tree-like continua." Fundamenta Mathematicae 155.2 (1998): 161-176. <http://eudml.org/doc/212249>.

@article{Hagopian1998,
abstract = {Let f be a map of a tree-like continuum M that sends each arc-component of M into itself. We prove that f has a fixed point. Hence every tree-like continuum has the fixed-point property for deformations (maps that are homotopic to the identity). This result answers a question of Bellamy. Our proof resembles an old argument of Brouwer involving uncountably many tangent curves. The curves used by Brouwer were originally defined by Peano. In place of these curves, we use rays that were originally defined by Borsuk.},
author = {Hagopian, Charles},
journal = {Fundamenta Mathematicae},
keywords = {fixed point; arc-component; deformation; tree-like continuum; Borsuk ray; dog-chases-rabbit argument; JFM 40.0372.03},
language = {eng},
number = {2},
pages = {161-176},
title = {The fixed-point property for deformations of tree-like continua},
url = {http://eudml.org/doc/212249},
volume = {155},
year = {1998},
}

TY - JOUR
AU - Hagopian, Charles
TI - The fixed-point property for deformations of tree-like continua
JO - Fundamenta Mathematicae
PY - 1998
VL - 155
IS - 2
SP - 161
EP - 176
AB - Let f be a map of a tree-like continuum M that sends each arc-component of M into itself. We prove that f has a fixed point. Hence every tree-like continuum has the fixed-point property for deformations (maps that are homotopic to the identity). This result answers a question of Bellamy. Our proof resembles an old argument of Brouwer involving uncountably many tangent curves. The curves used by Brouwer were originally defined by Peano. In place of these curves, we use rays that were originally defined by Borsuk.
LA - eng
KW - fixed point; arc-component; deformation; tree-like continuum; Borsuk ray; dog-chases-rabbit argument; JFM 40.0372.03
UR - http://eudml.org/doc/212249
ER -

References

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