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

Fundamenta Mathematicae (1998)

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

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## 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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1. [A] M. A. Armstrong, Basic Topology, McGraw-Hill, London, 1979.
2. [B] D. P. Bellamy, A tree-like continuum without the fixed point property, Houston J. Math. 6 (1979), 1-13. Zbl0447.54039
3. [Be] R. Bennett, Locally connected 2-cell and 2-sphere-like continua, Proc. Amer. Math. Soc. 17 (1966), 674-681. Zbl0139.40702
4. [Bi1] R. H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653-663. Zbl0043.16804
5. [Bi2] R. H. Bing, The elusive fixed point property, Amer. Math. Monthly 76 (1969), 119-132. Zbl0174.25902
6. [Bo1] K. Borsuk, Sur un continu acyclique qui se laisse transformer topologiquement en lui même sans points invariants, Fund. Math. 24 (1935), 51-58. Zbl0010.13402
7. [Bo2] K. Borsuk, A theorem on fixed points, Bull. Acad. Polon. Sci. 2 (1954), 17-20. Zbl0057.39103
8. [Br] L. E. J. Brouwer, On continuous vector distributions on surfaces, Proc. Konink. Akad. Wetensch. (Amsterdam) 11 (1909), 850-858.
9. [C] R. W. Conn, The engineering of magnetic fusion reactors, Scientific American 249 (4) (October, 1983), 60-71.
10. [Co] H. Cook, Tree-likeness of dendroids and λ-dendroids, Fund. Math. 68 (1970), 19-22. Zbl0203.25102
11. [H1] C. L. Hagopian, Fixed-point problems for disk-like continua, Amer. Math. Monthly 83 (1976), 471-473. Zbl0337.54027
12. [H2] C. L. Hagopian, Uniquely arcwise connected plane continua have the fixed-point property, Trans. Amer. Math. Soc. 248 (1979), 85-104. Zbl0407.54028
13. [H3] C. L. Hagopian, The fixed-point property for deformations of uniquely arcwise connected continua, Topology Appl. 24 (1986), 207-212. Zbl0606.54029
14. [H4] C. L. Hagopian, Fixed points of arc-component-preserving maps, Trans. Amer. Math. Soc. 306 (1988), 411-420. Zbl0642.54027
15. [H5] C. L. Hagopian, Fixed points of tree-like continua, in: Contemp. Math. 72, Amer. Math. Soc., 1988, 131-137.
16. [H6] C. L. Hagopian, A fixed-point theorem for tree-like continua, Topology Proc. 16 (1991), 57-62. Zbl0786.54043
17. [H7] C. L. Hagopian, Fixed-point problems in continuum theory, in: Contemp. Math. 117, Amer. Math. Soc., 1991, 79-86. Zbl0738.54011
18. [H8] C. L. Hagopian, The fixed-point property for simply-connected plane continua, Trans. Amer. Math. Soc. 348 (1996), 4525-4548. Zbl0878.54026
19. [L] S. Lefschetz, Continuous transformations of manifolds, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 90-93. Zbl49.0409.01
20. [Le] I. W. Lewis, Continuum theory problems, Topology Proc. 8 (1983), 361-394.
21. [M] H. F. Mathis, A short proof that an isotropic antenna is impossible, Proc. Institute Radio Engineers 39 (1951), 970.
22. [Mi1] P. Minc, A tree-like continuum admitting fixed point free maps with arbitrarily small trajectories, Topology Appl. 46 (1992), 99-106. Zbl0770.54043
23. [Mi2] P. Minc, A periodic point free homeomorphism of a tree-like continuum, Trans. Amer. Math. Soc. 348 (1996), 1487-1519. Zbl0863.54027
24. [OR] L. G. Oversteegen and J. T. Rogers, Jr., Fixed-point-free maps on tree-like continua, Topology Appl. 13 (1982), 85-95. Zbl0478.54028
25. [W] G. T. Whyburn, Analytic Topology, rev. ed., Amer. Math. Soc. Colloq. Publ. 28, Amer. Math. Soc., Providence, R.I., 1963. Zbl0117.15804
26. [Y] G. S. Young, Fixed-point theorems for arcwise connected continua, Proc. Amer. Math. Soc. 11 (1960), 880-884. Zbl0102.37806

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